In each of Problems 18 through 22,3"single power series whose generic tIn each of Proble ms 7 through 13, determine the Taylor series aboutthe point xo for the given function. Also determine the radius ofconvergence of the series.0018. n(n-1) a,x-2n -27. sin x,19. xΣna,x"-1 Σ αx+Σαπ'n=1k-08. ex, xo = 0m(m - 1)am"-220.+ x9. x, xo = lm=2k=10. x2, xo = -121. nanx"-1 +xanxn=011. Inx, xo = 1n=122. xn(n-1)a,x"-2+112.1- xn=0=223.Determine the a, so that the113.1 - x+nan"-1n=114. Let y nx".is satisfied. Try to identify the fn=0a. Compute y' and write out the first four terms of the series.uand write out the first four terms of the series.Σaχ.E-E

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Asked Nov 25, 2019
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How do I appropriately answer number 13? This is a question from my Differential Equations course, and the name of the chapter is, "Review of Power Series." I think that to determine the radius of convergence is by doing some kind of root test, ratio test, etc.

In each of Problems 18 through 22,
3"
single power series whose generic t
In each of Proble ms 7 through 13, determine the Taylor series about
the point xo for the given function. Also determine the radius of
convergence of the series.
00
18. n(n-1) a,x-2
n -2
7. sin x,
19. xΣna,x"-1 Σ αx
+Σαπ'
n=1
k-0
8. ex, xo = 0
m(m - 1)am"-2
20.
+ x
9. x, xo = l
m=2
k=
10. x2, xo = -1
21. nanx"-1 +xanx
n=0
11. Inx, xo = 1
n=1
22. xn(n-1)a,x"-2+
1
12.
1- x
n=0
=2
23.
Determine the a, so that the
1
13.
1 - x
+
nan"-1
n=1
14. Let y nx".
is satisfied. Try to identify the f
n=0
a. Compute y' and write out the first four terms of the series.
uand write out the first four terms of the series.
Σaχ.
E-E
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In each of Problems 18 through 22, 3" single power series whose generic t In each of Proble ms 7 through 13, determine the Taylor series about the point xo for the given function. Also determine the radius of convergence of the series. 00 18. n(n-1) a,x-2 n -2 7. sin x, 19. xΣna,x"-1 Σ αx +Σαπ' n=1 k-0 8. ex, xo = 0 m(m - 1)am"-2 20. + x 9. x, xo = l m=2 k= 10. x2, xo = -1 21. nanx"-1 +xanx n=0 11. Inx, xo = 1 n=1 22. xn(n-1)a,x"-2+ 1 12. 1- x n=0 =2 23. Determine the a, so that the 1 13. 1 - x + nan"-1 n=1 14. Let y nx". is satisfied. Try to identify the f n=0 a. Compute y' and write out the first four terms of the series. uand write out the first four terms of the series. Σaχ. E-E

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Expert Answer

Step 1

We are given the function 1/(1-x). In this problem we need to find the Taylor series about the point x0 = 2. Also, we need to find the radius of convergence of the series formed.

Note:

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The Taylor series of a real valued function f(x) that is infinitely differentiable at a real number a is the power series "(a) (x- a)2 f'(a) f'(a) (x - a) (х — а)3 + .. f(a) 1! 2! 3! In the more compact notation, 00 'f) (a) п! n-0

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Step 2

Here,

...
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1 f(x) 1- x 1 f'(x) (1 — х)2 2 f"(x) (1 — х)3 6 f'"'(x) (1 — х)4 So, the Taylor series about the point xo- 2 is, f'(2) f'"'(2) f"(2) (x- 2)2 +(x2)3 6(х — 2)3 (x 2) (x 2) In the compact form, the series becomes, 0(1)+1 n!(x 2)" f(2) 1! 2! 3! 2(х — 2)2 -1 + .. --(1)

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