(ii) Apply the ratio test:
if
l=lim
n→∞∣
∣
∣
∣un+ 1
un∣
∣
∣
∣then the series converges ifl<1 and diverges ifl>1.
(iii) Ifl=1, use a comparison test with a known standard series.
(iv) If the series is alternating then it is convergent ifun→0, asn→∞.Exercise on 14.10
Test the following series for convergence
(i)1
2+1
3(
1
2) 3
+1
5(
1
2) 5
+1
7(
1
2) 7
+···+1
2 r− 1(
1
2) 2 r− 1
+···(ii)1
2+1
3(
−1
2) 3
+1
5(
−1
2) 5
+1
7(
−1
2) 7
+···+1
2 r− 1(
−1
2) 2 r− 1
···(iii) 2+
1
323 +1
525 +···+1
2 n− 122 n−^1 +···Answer
(i) C (ii) C (iii) D
14.11 Infinite power series
Aseriesofform
a 0 +a 1 x+a 2 x^2 +a 3 x^3 +···where the coefficients are constants, is called aninfinite power series inx(107
➤
).
Such series are used to provide useful alternative forms for functions – for example
ex= 1 +x+x^2
2!+x^3
3!+···cosx= 1 −x^2
2!+x^4
4!−··· (xin radians)The right-hand sides have the advantage that they can, subject to convergence, be used
to evaluate the function to any required accuracy. In general such series are obviously
most helpful for small values ofx. We say they give approximationsnear to the origin
x=0 and they are also calledpower series about the origin,orMaclaurin’s series.If
we require series that approximate a function near to some particular non – zero value of
x,sayx=a,thenweuseaTaylor series aboutx=a:
f(x)=a 0 +a 1 (x−a)+a 2 (x−a)^2 +···