Mathematics: Find a power series representation for f(x) = 1 (10 + x)2 . f(x) = ∞ n = 0 what is the radius of convergence, r? r = (b) use part (a) to find a power series representation for f(x) = x3 (10 + x)2 . f(x) = ∞ n = 0 what is the radius of convergence, r?

Find a power series representation for f(x) = 1

Ответ:

icantspeakengles

15.03.2022

Mathematics

a. Recall that for |x| , we have

$\dfrac1{1-x}=\displaystyle\sum_{n\ge0}x^n$

which has derivative

$\dfrac1{(1-x)^2}=\displaystyle\sum_{n\ge0}nx^{n-1}=\sum_{n\ge0}(n+1)x^n$

Then

$f(x)=\dfrac1{(10+x)^2}=\dfrac1{100}\dfrac1{\left(1-\left(-\frac x{10}\right)\right)^2}=\frac1{100}\displaystyle\sum_{n\ge0}(n+1)\left(-\frac x{10}\right)^n$

which converges

$\left|-\dfrac x{10}\right|$

b. From the above result, it's evident that the radius of convergence is R=10 .

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Ответ:

Esme1995

04.03.2023

Mathematics

have attached the solution, couldn't solve 13 and 17 tho, if it helps, then do mark me the brainliest...

Btw, which class r u in?

Step-by-step explanation:

Please answer i will give you brainlest please help me

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Find a power series representation for f(x) = 1 (10 + x)2 . f(x) = ∞ n = 0 what is the radius of convergence, r? r = (b) use part (a) to find a power series representation for f(x) = x3 (10 + x)2 . f(x) = ∞ n = 0 what is the radius of convergence, r?

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