- an infinite sum of, for example, the form
- ul + u2 + u3 + … + un
*+*… - or, more concisely, (1)

- A simple example of an infinite series encountered in elementary mathematics is the sum of a decreasing geometric progression:
- (2) 1
*+ q + q2 +*…*+ (qn +*…*=*1/(1 –*q) ǀqǀ*< 1 - Often called simply series, infinite series are extensively used in mathematics and its applications both in theoretical studies and in approximate numerical solutions of problems.
- Many numbers can be expressed in the form of special infinite series that permit easy calculation of the approximate values of the numbers to the required degree of accuracy. For example, the number πcan be computed from the series

For the base *e* of natural logarithms there exists the series

The value of the natural logarithm ln 2 can be obtained from the series

Expansion in infinite series is a powerful technique for studying functions. Series expansions are used, for example, to calculate approximate values of functions, to calculate and estimate the values of integrals, and to solve algebraic, differential, and integral equations.

When in numerical calculations we replace an infinite series by the sum of the initial terms of the series, it is useful to have an estimate of the resultant error. This gives us an estimate of the rate of convergence of the series.

Moreover, it is desirable to use series for which these errors tend to zero rapidly as the number *n* of terms increases.

For example, in the case of series (4) the error estimate has the form 0 < e - sn* 0, there exist a number n∊ such that*

- for all
*n > n*∊ and all integers*ρ*≥ 0. It follows that if series (1) converges, then - The converse is not true: the
*nth*term of the harmonic series

tends toward zero, although this series diverges.

Series with nonnegative terms play an important role in the theory of series. In order for such a series to converge, it is necessary and sufficient that the sequence of its partial sums be bounded above. If, however, the series diverges, then

- We therefore write in this case
- A number of tests for convergence exist for series with non-negative terms.
- According to the integral test for convergence, if the function
*f(x*) is defined for all*x*≥ 1, is nonnegative, and decreasing, then the series

- converges if, and only if, the integral
- converges. Through the use of this test we can easily establish that the series

converges when α *>* 1 and diverges when α ≤ 1.

Convergence can also be verified by the comparison test: if for any two series (1) and (6) with nonnegative terms there exists a constant *c >* 0 such that 0 ≤ un ≤ cvn, then the convergence of series (6) implies the convergence of series (1), and the divergence of series (1) implies the divergence of series (6). Series (8) is usually selected for comparison, and in the given series the principal part of the form *A/n*α is singled out. By using this method we can immediately demonstrate the convergence of the series with the nth term

- where
- This series converges because the series
- converges.
- The following rule is a corollary of the comparison test: if
- then the series converges when α > 1 and 0 ≤
*k*1. The other is Cauchy’s root test: if

exists, then series (1) converges when 1 < 1 and diverges when 1 > 1. Both in the case of d’ Alembert’s test and in the case of Cauchy’s test the series can be divergent or convergent when 1 = 1.

- Absolutely convergent series constitute an important class of infinite series. Series (1) is said to be absolutely convergent if the series
- converges.
- If a series converges absolutely, then it also converges in the ordinary sense. The series
- converges absolutely; the series

however, converges only in the ordinary sense. The sum of absolutely convergent series and the product of an absolutely convergent series and a number are also absolutely convergent series. The properties of finite sums are carried over most completely to absolutely convergent series. Suppose

is a series whose terms are the same as those of series (1) but are in general arranged in a different order. If series (1) converges absolutely, then series (9) also converges and has the same sum as series (1).

If series (1) and (6) converge absolutely, then the series obtained from all possible products umvn of these series’ terms, arranged in an arbitrary order, also converges absolutely.

Moreover, if the sum of the series is equal to

sand the sums of series (1) and (6) are equal to s1 and s2, respectively, then s = s1s2; in other words, absolutely convergent series can be multiplied term by term without concern for the order of the terms. The convergence test for series with nonnegative terms can be used to establish the absolute convergence of series.

Convergent series that do not converge absolutely are said to be conditionally convergent. The sums of such series are not independent of the order of the terms. According to Riemann’s theorem, by an appropriate rearrangement of the terms of a given conditionally convergent series we can obtain a divergent series or a series that has a prescribed sum. The series

- is an example of a conditionally convergent series. If the terms of this series are rearranged so that two positive terms are followed by one negative term

then the sum of the series will be increased by a factor of 1.5. Several tests for convergence exist that are applicable to conditionally convergent series. An example is Leibniz’ test: if

- then the alternating series
- converges. Through the use of, for example, Abel’s transformation more general tests can be obtained for series representable in the form
- One such test is Abel’s test: if the sequence {an} is monotonie and bounded and the series
- converges, then series (11) also converges. According to Dirichlet’s test, if the sequence {an} tends monotonically to zero and the sequence of the partial sums of the series
- is bounded, then series (11) converges. For example, by applying Dirichlet’s test we can show that the series
- converges for all real α.
- Series of the form
- are sometimes considered. Such a series is said to be convergent if the series

converge. The sum of these series is called the sum of the original series.

- Multiple series have a more complex structure. These are series of the form
- where the

*un1, n2,…, nk*

are given—in general, complex—numbers indexed by *k* subscripts n1, n2, …, n2, each of which varies independently over the natural numbers. The simplest series of this type are double series.

For some series of numbers it is possible to obtain simple formulas for the value of the remainder or for an estimate of the remainder. These formulas are extremely important, for example, in evaluating the accuracy of calculations made with series. Thus, the remainder for the sum of geometric progression (2) is

*rn*=*qn+1/(a – q) ǀqǀ < 1*- For series (7), under the assumptions made, we have
- For series (10) we have the formula
- ǀ
*rn*ǀ ≤*un+1*

By using certain special transformations we can sometimes improve the convergence of a convergent series. Both convergent series and divergent series are used in mathematics. In the case of divergent series, more general concepts of the sum of a series are introduced. For example, divergent series (5) can be summed in a certain way to ½.

** Series of functions.** The concept of an infinite series can be extended in a natural way to the case where the terms of the series are the functions un = un (

*x*) defined on some set

*E;*these can be real functions, complex functions, or, more generally, functions whose values belong to some metric space. The series

- in this case is said to be a series of functions.
- If series (1’) converges at every point of E, then it is said to be convergent on
*E.*Thus the series

converges throughout the complex plane. The sum of a convergent series of functions that are continuous, for example, on some closed interval is not necessarily a continuous function.

The properties of continuity, differentiability, and integrability of finite sums of functions carry over to series of functions under certain conditions that are formulated in terms of uniform convergence of series.

Convergent series (1’) is said to be uniformly convergent on *E* if, for all points in *E* and for sufficiently large n, the deviation of the partial sums of the series

- from the sum of the series
- does not exceed the same arbitrarily small quantity. More precisely, the series is uniformly convergent if, for arbitrary ∊ > 0, there exists an
*n*∊ such that - ǀ
*s(x) – sn(x)ǀ ≤ ∊* - for all n ≥ n∊ and for all points x ∊
*E.*This condition is equivalent to - where
*r*n(*x*) =*s*(*x*) –*s*n(*x*) and - is the least upper bound of Irn (
*x*) I on*E.*For example, the series - converges uniformly on the closed interval [0,
*q]*when 0 <*q*0, there exists an*n*∊ such that - ǀ
*u*(*x*) + … +*u*n+p(*x*ǀ < ∊ - for all
*n ≥ n*∊ all*ρ*≥ 0, and all points*x*∊*E.* - According to Weierstrass’ test, if there exists a convergent series
- such that ǀun(
*x*)ǀ ≤ an*, x*∊*E, n*= 1, 2, …, then series (1’) converges uniformly on*E.*

The sum of a uniformly convergent series of functions that are continuous on some closed interval (or, more generally, on some topological space) is a continuous function on this interval (on this space).

The sum of a uniformly convergent series of functions integrable on some set is an integrable function on this set, and the series can be integrated term by term.

If the sequence of partial sums of a series of integrable functions converges in the mean to some integrable function, then the integral of this function is equal to the sum of the series of integrals of the terms of the series. Integrability here is construed in the sense of Riemann or Lebesgue.

For Lebesgue integrable functions, a series with an almost everywhere convergent sequence of partial sums can be integrated term by term if we have a uniform estimate of the partial sums’ absolute values by some Lebesgue integrable function.

Suppose series (1’) is convergent on some closed interval; if the terms of the series are difierentiable on the interval and if the series of their derivatives converges uniformly, then the sum of the original series is also differentible on this interval, and the series can be differentiated term by term.

The concept of a series of functions can also be generalized to the case of multiple series. In different branches of mathematics and its applications, extensive use is made of the expansions of functions in series of functions, especially in power series, trigonometric series, and, more generally, series of eigenfunctions of some operators.

*History.*

## Series

A **series** is just the sum of some set of terms of a sequence. For example, the sequence 2, 4, 6, 8, … has partial sums of 2, 6, 12, 20, … These partial sums are each a **finite series**.

The nth partial sum of a sequence is usually called Sn. If the sequence being summed is sn we can use sigma notation to define the series: which just says to sum up the first n terms of the sequence s.

If we sum up an infinite number of terms of a sequence we get an **infinite series**.

This device cannot display Java animations. The above is a substitute static image

See About the calculus applets for operating instructions. |

### 1. Arithmetic series

s1 and d set to generate odd integers. On this applet, the sequence is shown as rectangles of width 1, somewhat reminiscent of a Riemann sum. The first term of this sequence is 1 so the first rectangle is 1 by 1 wide. The second term in our example is 3 so the next rectangle is 3 high by 1 wide, the third term is 5 high by 1 wide and so on.

The dots on this graph represent the different finite series, each being the sum of the terms of the sequence up to that point. From a visualization standpoint, think of the height of each dot as being the total area of all the rectangles to the left of the dot.

Hence the first dot is at (1,1), the second dot is at (2,3), the third is at (3,5), etc. Input boxes and sliders are provided to allow you to change s1 and d.

The table on the left gives some large values of the sequence and the series, clearly showing that the infinite series diverges.

## Sequences | Brilliant Math & Science Wiki

A **sequence** is an ordered set with members called terms.

Usually, the terms are numbers. A sequence can have infinite terms.

An example of a sequence is

1,2,3,4,5,6,7,8,… .1,2,3,4,5,6,7,8,dots.1,2,3,4,5,6,7,8,….

There are different types of sequences. For example, an arithmetic sequence is when the difference between any two consecutive terms in the sequence is the same. So,

5,14,23,32,41,505, 14, 23, 32, 41,505,14,23,32,41,50

is an arithmetic sequence with common difference 999, first term 555, and number of terms 6.6.6.

Another type of sequence is a geometric sequence. This is when the ratio of any two consecutive terms in the sequence is the same. For example,

2,6,18,54,1622, 6, 18, 54, 1622,6,18,54,162

is a geometric sequence with common ratio 333, first term 222, and number of terms 5.5.5.

In a sequence, it is conventional to use the following variables:

- aaa is the first term in the sequence.
- nnn is the number of terms in the sequence.
- Tn{ T }_{ n }Tn is the nthn^ ext{th}nth term in the sequence.
- Sn{ S }_{ n }Sn is the sum of the first nnn terms of the sequence.
- ddd is the common difference between any two consecutive terms (arithmetic sequences only).
- rrr is the common ratio between any two consecutive terms (geometric sequence only).

For example, if a series starts with 111 and has a common difference of 1,1,1, we have Sn=n(n+1)2.{ S }_{ n }= dfrac{n(n + 1)}{2}.Sn=2n(n+1).

Similarly, for the series of squares 12,22,32,…,n2,1^2,2^2,3^2,dots,n^2,12,22,32,…,n2, we have Sn=n(n+1)(2n+1)6.{ S }_{ n } = dfrac{n(n + 1)(2n + 1)}{6}.Sn=6n(n+1)(2n+1).

For the series of cubes 13,23,33,…,n3,1^3,2^3,3^3,dots,n^3,13,23,33,…,n3, we have Sn=(n(n+1)2)2.{ S }_{ n } = left(dfrac{n(n+1)}{2}

ight)^2.Sn=(2n(n+1))2.

Some special types of sequences can be found in Arithmetic Progressions, Geometric Progressions, and Harmonic Progression.

## Finite and Infinite Mathematical Series

**Overview: What Is A Mathematical Series?**

A mathematical series is the sum of the elements of a mathematical sequence. If the series is finite, the sum will be a finite number. For example, in the finite sequence {1, 2, 3, 4, 5}, the series is the sum of all the terms: 1 + 2 +3 +4 + 5 = 15.

**Partial Sums and Sigma**

Partial sums can be used, if the series is very long. The final sum in the above example is easy to find, but it can also be approached by finding partial sums, such as the third partial sum in the series, S3, which is the sum of all the values of the first three numbers in the sequence, 1 +2 +3 or 6. In formula terms, directions to find the sum are given by a Greek letter, sigma or ∑.

**Infinite Mathematical Series**

What is the mathematical sequence is infinite? Finding the sum isn’t as easy as in a finite series, and the directions to find partial sums are very important.

For example, in the series {2 + 4 +6 +… + 2n …} , the sum of the first term , S1=2, the second partial sum, S2, is 2 +4, or 6, and the third partial sum, S3, is 2 +4 +6 = 12.

The series will go on infinitely, because for every 2n, the nth term can always be one number larger.

**Are All Mathematical Series Infinite?**

The sum of many mathematical sequences is an infinitely large number, because n is infinitely large.

However, some mathematical sequences are infinite, but n becomes infinitely small, and the sum approaches a limit, For example, suppose the mathematical sequence starts with a square with area 1, and then divide the square into halves.

The first rectangle will have an area 1/2 the size of the original square, the second rectangle will have an area 1/4 the size, the third will have an area 1/8, the fourth will have the area 1/16, and so on.

The pattern can be expressed as (1/2)1, (1/2)2, (1/2)3…(1/2)n… If it is written as a series , then 1/2 +1/4 +1/8 +1/16 +… = 1, because the original square has area 1 and it was just divided into n rectangles. The infinite series has a finite limit.

**Real-World Applications Of Mathematical Series**

Mathematical series have a number of useful applications. For example, determining how much money will be earned with a recurring investment is an application of mathematical series used in banking. There are a number of mathematical series that are used in radio and electronics, as well as in physics and computer science.

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## Series – Mathematics A-Level Revision

The series of a sequence is the sum of the sequence to a certain number of terms. It is often written as Sn. So if the sequence is 2, 4, 6, 8, 10, … , the sum to 3 terms = S3 = 2 + 4 + 6 = 12.

**The Sigma Notation**

The Greek capital sigma, written S, is usually used to represent the sum of a sequence. This is best explained using an example:

- This sum is therefore equal to 3×1 + 3×2 + 3×3 + 3×4 = 3 + 6 + 9 + 12 = 30.
- 3 S 3r + 2
- r = 1
- This is equal to: (3×1 + 2) + (3×2 + 2) + (3×3 + 2) = 24 .
**The General Case**- n S Ur r = 1

This is the general case. For the sequence Ur, this means the sum of the terms obtained by substituting in 1, 2, 3,… up to and including n in turn for r in Ur. In the above example, Ur = 3r + 2 and n = 3.

**Arithmetic Progressions**

An arithmetic progression is a sequence where each term is a certain number larger than the previous term. The terms in the sequence are said to increase by a common difference, d.

For example: 3, 5, 7, 9, 11, is an arithmetic progression where d = 2. The nth term of this sequence is 2n + 1 .

In general, the nth term of an arithmetic progression, with first term a and common difference d, is: a + (n – 1)d . So for the sequence 3, 5, 7, 9, … Un = 3 + 2(n – 1) = 2n + 1, which we already knew.

**The sum to n terms of an arithmetic progression**

This is given by:

- Sn = ½ n [ 2a + (n – 1)d ]

- You may need to be able to prove this formula. It is derived as follows:
- The sum to n terms is given by: Sn = a + (a + d) + (a + 2d) + … + (a + (n – 1)d) (1)
- If we write this out backwards, we get: Sn = (a + (n – 1)d) + (a + (n – 2)d) + … + a (2)
- Now let’s add (1) and (2): 2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + … + [2a + (n – 1)d] So Sn = ½ n [2a + (n – 1)d]
**Example**

Sum the first 20 terms of the sequence: 1, 3, 5, 7, 9, … (i.e. the first 20 odd numbers).

- S20 = ½ (20) [ 2 × 1 + (20 – 1)×2 ] = 10[ 2 + 19 × 2] = 10[ 40 ]
- = 400
**Geometric Progressions**

A geometric progression is a sequence where each term is r times larger than the previous term. r is known as the common ratio of the sequence. The nth term of a geometric progression, where a is the first term and r is the common ratio, is:

For example, in the following geometric progression, the first term is 1, and the common ratio is 2: 1, 2, 4, 8, 16, …

- The nth term is therefore 2n-1
**The sum of a geometric progression**- The sum of the first n terms of a geometric progression is:
- We can prove this as follows:
- Sn = a + ar + ar2 + … + arn-1 (1)
- Multiplying by r: rSn = ar + ar2 + … + arn (2)
- (1) – (2) gives us: Sn(1 – r) = a – arn (since all the other terms cancel)
- And so we get the formula above if we divide through by 1 – r .
**Example**- What is the sum of the first 5 terms of the following geometric progression: 2, 4, 8, 16, 32 ? S5 = 2( 1 – 25) 1 – 2
- = 2( 1 – 32)
- = 62
**The sum to infinity of a geometric progression**

-1

In geometric progressions where |r| < 1 (in other words where r is less than 1 and greater than –1), the sum of the sequence as n tends to infinity approaches a value. In other words, if you keep adding together the terms of the sequence forever, you will get a finite value. This value is equal to:

**Example**

Find the sum to infinity of the following sequence:

1 | , | 1 | , | 1 | , | 1 | , | 1 | , | 1 | , | … |

2 | 4 | 8 | 16 | 32 | 64 |

Here, a = 1/2 and r = 1/2

Therefore, the sum to infinity is 0.5/0.5 = 1 .

So every time you add another term to the above sequence, the result gets closer and closer to 1.

**Harder Example**

The first, second and fifth terms of an arithmetic progression are the first three terms of a geometric progression. The third term of the arithmetic progression is 5. Find the 2 possible values for the fourth term of the geometric progression.

- The first term of the arithmetic progression is: a The second term is: a + d The fifth term is: a + 4d
- So the first three terms of the geometric progression are a, a + d and a + 4d .

In a geometric progression, there is a common ratio. So the ratio of the second term to the first term is equal to the ratio of the third term to the second term. So:a + d = a + 4d a a + d (a + d)(a + d) = a(a + 4d) a² + 2ad + d² = a² + 4ad d² – 2ad = 0 d(d – 2a) = 0

- therefore d = 0 or d = 2a
- The common ratio of the geometric progression, r, is equal to (a + d)/a Therefore, if d = 0, r = 1 If d = 2a, r = 3a/a = 3
- So the common ratio of the geometric progression is either 1 or 3 .

We are told that the third term of the arithmetic progression is 5. So a + 2d = 5 . Therefore, when d = 0, a = 5 and when d = 2a, a = 1 . So the first term of the arithmetic progression (which is equal to the first term of the geometric progression) is either 5 or 1.

Therefore, when d = 0, a = 5 and r = 1. In this case, the geometric progression is 5, 5, 5, 5, …. and so the fourth term is 5.When d = 2a, r = 3 and a = 1, so the geometric progression is 1, 3, 9, 27, … and so the fourth term is 27.

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