MATH 106A,B - CALCULUS II WINTER 2008
QUIZ 10
NAME:
Show ALL your work CAREFULLY.
(a) Find the interval of convergence of the following infinite series [Do not forget to check the endpoints!].
โ
X
k=1
(4x)k
k2
Consider
lim
kโโ
|4x|k+1
(k+ 1)2ยทk2
|4x|k= lim
kโโ
|4x|๎k
k+ 1๎2
=|4x|lim
kโโ ๎k
k+ 1๎2
=|4x|.
By the Ratio Test, the series converges if |4x|<1, i.e., โ1<4x < 1or โ1
4<x< 1
4. When
x=1
4, the series becomes P1
k2which is known to converge. When x=โ1
4, the series becomes
P(โ1)k
k2which converges since the associated series of absolute values P1
k2converges. Hence,
the interval of convergence is [โ1
4,1
4].
(b) Recall that 1
1โt= 1 + t+t2+t3+t4+... for |t|<1.
Use this to obtain a power series representation of arctanx. [What is the derivative of arctanx?]
By letting t=โx2, we have
1
1 + x2= 1 โx2+x4โx6+... for |x|<1.
It follows that
arctan x=Z1
1 + x2dx =xโx3
3+x5
5โx7
7+...
=X
n=1
(โ1)n+1x2nโ1
2nโ1for |x|<1.
(c) Use (b) to evaluate limxโ0arctanx
x.
Using the power series representation of arctan xfrom (b), we have
lim
xโ0
arctan x
x= lim
xโ0
1
x๎xโx3
3+x5
5โx7
7+...๎
= lim
xโ0๎1โx2
3+x4
5โx6
7+...๎= 1.
Date: April 4, 2008.
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