Use the power series To determine a power series centered at 0, for the function.Identify the interval of convergence. f(x)=ln(x^2+1)

Ibrahim Rosales 2022-07-28 Answered
Use the power series
To determine a power series centered at 0, for the function.Identify the interval of convergence.
f ( x ) = ln ( x 2 + 1 )
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Answers (2)

Jaycee Figueroa
Answered 2022-07-29 Author has 10 answers
g ( x ) = 1 1 + x = 1 x + x 2 x 3 + x 4
We must recognize a connection between 1/(1+x) and the naturallog. well if we integrate what we have now we'd end up with ln ( 1 + x ), so that's a start, but we want to end up with ln ( x 2 + 1 ). So we'd need 2 x / [ 1 + x 2 ]. So we have
g ( x 2 ) = 1 1 + x 2 = 1 x 2 + x 4 x 6 + x 8
Multiplying everything by x
x g ( x 2 ) = x 1 + x 2 = x x 3 + x 5 x 7 + x 9
And by 2
2 x g ( x 2 ) = 2 x 1 + x 2 = 2 x 2 x 3 + 2 x 5 2 x 7 + 2 x 9
Integrating letting u = 1 + x 2 d u = 2 x d x, we find
ln ( 1 + x 2 ) = x 2 x 4 2 + x 6 3 x 8 4 + x 10 5
Hence
ln ( 1 + x 2 ) = n = 1 ( 1 ) n + 1 x 2 n n
lim n | x 2 n + 1 n + 1 n x 2 n | < 1
| x | lim n | n n + 1 | < 1
|x|<1
Now we must check the endpoints individually. For x =1, we have a alternating harmonic series, which isconvergent. Same for x = -1. So the intervalis [-1,1]
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Greyson Landry
Answered 2022-07-30 Author has 5 answers
1 x + x 2 x 3 + x 4 x 5 + x 6 x 7 + x 8 x 9 + x 10 x 11 + x 12 x 13 + x 14 x 15 + . . . . . . . . . . . . . . . . . . . .
f ( x ) = ln ( x 2 + 1 ) = x 2 x 4 / 2 + x 6 / 3 x 8 / 4 + x 10 / 5 x 12 / 6 + . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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