Understanding Engineering Mathematics

So the ratio test tells us that this series converges for all finitex. It is, of course, the

series forex.

(ii) Forx−

x^2

2

+

x^3

3

−

x^4

4

+···,wehave|un|=

∣

∣

∣

∣

xn

n

∣

∣

∣

∣and so

lim

n→∞

∣

∣

∣

∣

un+ 1

un

∣

∣

∣

∣=nlim→∞

∣

∣

∣

∣

xn+^1

(n+ 1 )

n

xn

∣

∣

∣

∣=nlim→∞

∣

∣

∣

∣

nx

n+ 1

∣

∣

∣

∣=|x|

So by the ratio test the series is convergent if|x|<1 and divergent if|x|>1. If

|x|=1 the ratio test is inconclusive and we have to consider this case separately.

There are two cases to consider,x=1andx=−1.

Ifx=1theseriesis

S= 1 −^12 +^13 −^14 +···

and this is an alternating series with decreasing terms that tend to zero, and so itconverges.
Ifx=−1:

S=− 1 −^12 −^13 −^14 +···

and this is the negative of the harmonic series and sodiverges.
Thus we can summarize our results:

x−

x^2

2

+

x^3

3

−···

converges if− 1 <x≤1 and diverges ifx>1orx≤−1. This is in fact the series for
log( 1 +x).
Note that the ratio test ensuresabsolute convergence, since it involves taking the
modulus of the term of the series. Thus the series for tan−^1 x(see Exercise on 14.11)
is absolutely convergent forx^2 <1 and in particular can be differentiated term by term
to give

1

1 +x^2

= 1 −x^2 +x^4 −x^6 +···

which can be checked by the binomial theorem (71

➤

).

Exercise on 14.11

Derive the series tan−^1 x=x−

x^3

3

+

x^5

5

−

x^7

7

+···+(− 1 )n+^1

x^2 n−^1

2 n− 1

+···and discuss

its convergence.

Answer

The Maclaurin’s series for tan−^1 xis valid for− 1 ≤x≤1.