Series, alternating series, power series, convergence

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SERIES, ALTERNATING SERIES, POWER SERIES, CONVERGENCE

Def. Series. The indicated sum of a finite or infinite sequence of terms. It is a finite or an infinite series according as the number of terms is finite or infinite. An infinite series can be written in the form

a₁ + a₂ + a₃ + ... + a_n + ...

or Σ a_n, where a_n is the general term or the n-th term. Infinite series is usually shortened to series, as in convergent series, Taylor’s series, etc. A series is a positive series if its terms are all positive real numbers and a negative series if they are all negative real numbers.

James and James. Mathematics Dictionary.

Def. Sum of an infinite series. An infinite series of numbers has no sum within the ordinary meaning of the word since no matter how many of the terms one might add, there would always be an unlimited number of them left over. We must therefore define a meaning for the term sum as applied to infinite series.

Consider a series

u₁ + u₂ + u₃ + ... + u_n + ...

and the sequence of partial sums S₁, S₂, S₃, ... where

S₁ = u₁

S₂ = u₁ + u₂

S₃ = u₁ + u₂ + u₃

...............................

S_n = u₁ + u₂ + u₃ + .... + u_n

If S_n, regarded as a function of n, approaches a limit S as n → ∞, this limit is called the sum of the infinite series and the series is said to be convergent. If S_n does not approach a limit as n → ∞, the series is said to be divergent.

Example. The sum of the series

is 1, because that is the limit approached by the sum of the first n of these terms, namely 1 - 1/2ⁿ, as n becomes infinite. The sum of the series is precisely 1, even though the actual arithmetic sum of a finite number of terms of the series is always less than 1. The series 1 + (-1) + 1 + (-1) + 1 + ... does not have a sum, since the sum of the first n terms is + 1 if n is odd and 0 if n is even and therefore the sum of the first n terms does not have a limit as n increases.

A series can converge in only one way: lim S_n as n → ∞ must exist (and be finite). But a series can diverge in two ways:

Example. The limit of the series

1 + (-1) + 1 + (-1) + 1 + ...

just simply fails to exist.

Middlemiss. Differential and Integral Calculus.

James and James. Mathematics Dictionary.

Theorem. Multiplication of each term of a series by a constant different from zero does not affect the convergence or divergence. Removal (or addition) of a finite number of terms from (or to) a series does not affect the convergence or divergence.

Def. Mixed-term series. A series that contains an unlimited number of both positive and negative terms.

Def. Alternating series. A series whose terms are alternately positive and negative, as

An alternating series converges if each term is numerically equal to or less than the preceding and if the n-th term approaches zero as n increases without limit. This is a sufficient, but not a necessary set of conditions — the term-by-term sum of any two convergent series converges and, if one series has all negative terms and the other all positive terms, their indicated sum may be a convergent alternating series and not have its terms monotonically decreasing. The series

is such a series.

James and James. Mathematics Dictionary.

Convergence and divergence of some important series.

Geometric series

a + ar + ar² + ... + arⁿ + ...

a and r are constants

Converges to

if | r | < 1

Diverges if | r | 1

Harmonic series

Diverges

Factorial series

Converges to “e”

p series

Converges if p > 1

Diverges if p 1

Binomial series

Converges if |x| < 1

Diverges if |x| > 1

Necessary condition for the convergence of an infinite series. A necessary condition for the convergence of an infinite series is that the terms approach zero as one goes further out in the series i.e. that the n-th term approaches zero as n becomes infinite. This is not a sufficient condition for convergence; e.g. the series

is divergent, although 1/n approaches zero as n becomes infinite.

James and James. Mathematics Dictionary.

Cauchy’s condition for the convergence of an infinite series. Cauchy’s condition for the convergence of an infinite series is that the sum of any number of terms can be made as small as desired by starting sufficiently far out in the series. Tech. A necessary and sufficient condition for convergence of an infinite series Σa_n is that, for any ε > 0 , there exists an N such that

| a_n + a_n+1+ ... + a_n+h| < ε

for all n > N and all h > 0.

James and James. Mathematics Dictionary.

Absolute and conditional convergence of an infinite series.

Theorem 1. A mixed-term series Σu_n will converge if the corresponding positive series Σ |u_n | obtained by replacing all terms of the series by their absolute values converges.

Def. Absolute convergence. A series Σu_n is said to converge absolutely (or to be absolutely convergent) if the corresponding positive series Σ |u_n | obtained by replacing all terms of the series by their absolute values converges.

Example. The series

is absolutely convergent since the series

converges.

Def. Conditional convergence. A series is called conditionally convergent if it is convergent, but not absolutely convergent.

Example. The series

is conditionally convergent since it converges but the series

diverges..

● If a given series of mixed terms is absolutely convergent then the positive terms alone form a convergent series and the negative terms form a convergent series. If the series of positive terms converges to P and the series of negative terms converges to - Q, then the given series converges to P - Q.

● If a given series is only conditionally convergent, then the positive terms alone form a divergent series and the negative terms form a divergent series.

● An absolutely convergent series remains absolutely convergent, and with the same sum, when the terms are arranged in any other order.

● A conditionally convergent series may become divergent, or may converge to a different sum, if the terms are taken in a different order. Thus the series

is merely a rearrangement of the series

It can be shown that the first series is convergent and its sum is half that of the second series. The terms of a conditionally convergent series may in fact be rearranged so as to make the value of the series any desired quantity.

Middlemiss. Differential and Integral Calculus.

Oakley. The Calculus. p. 188

Addition of infinite series. Addition of infinite series is performed by adding corresponding terms i.e. the sum of two infinite series is defined as the series obtained by adding the corresponding terms of the two series. Thus the sum of the series

a₁ + a₂ + a₃ + .... + a_n + ...

and

b₁ + b₂ + b₃ + .... + b_n + ... ,

is the series

(a₁ + b₁) + (a₂ + b₂) + (a₃ + b₃) + .... + (a_n + b_n) + ... .

If two convergent series of constant terms,

a₁ + a₂ + a₃ + .... + a_n + ...

and

b₁ + b₂ + b₃ + .... + b_n + ... ,

have sums S and S', then the series

(a₁ + b₁) + (a₂ + b₂) + (a₃ + b₃) + .... + (a_n + b_n) + ...

converges and has the sum S + S'. If the series

u₁ + u₂ + u₃ + .... + u_n + ...

and

v₁ + v₂ + v₃ + .... + v_n + ... ,

whose terms are functions of x, converge in certain intervals, the term by term sum of these series, namely

(u₁ + v₁) + (u₂ + v₂) + (u₃ + v₃) + .... + (u_n + v_n) + ...

converges in any interval common to the two intervals.

James and James. Mathematics Dictionary.

Multiplication of infinite series. If Σa_n and Σb_n are any two series, their product is defined as the series Σc_n

Σc_n = a₁b₁ + (a₁b₂ + a₂b₁) + (a₁b₃ + a₂b₂ +a₃b₁) + ... + (a₁b_n + a₂b_n-1+ ... + a_nb₁) + ...

If one multiplies together two convergent series having sums U and V, respectively, the resulting series will converge with sum U V if at least one of the series is absolutely convergent. If both series are only conditionally convergent the product series may be divergent.

A power series converges absolutely within its interval of convergence; hence two power series can always be multiplied, and the result will be valid within their common interval of convergence.

Def. Power series. A series whose terms contain ascending positive integral powers of a variable, a series of the form

a₀ + a₁x + a₂x² + ... + a_nxⁿ + ...

where the a’s are constants and x is a variable; or a series of the form

a₀ + a₁(x - h) + a₂(x - h)² + ... + a_n(x - h)ⁿ + ... ,

James and James. Mathematics Dictionary.

Approximation of the sum of an infinite series and estimation of error. Suppose a series Σu_n converges to sum S. Suppose we wish to approximate its sum by using the partial sum S_k of the first k terms. We need to have some idea of the maximum error we incur by using only the first k terms. Let

R_k = S - S_k.

R_k then is the remainder after k terms. We wish estimates of R_k..

1. Convergent alternating series. For a convergent alternating series,

|R_k| < u_k+1

i.e. R_k is less than the first neglected term. Moreover, R_k is positive when k is even and negative when k is odd.

2. Convergent geometric series. For the convergent geometric series Σarⁿ,