(h). Use your equation in part (g) to compute the value of \( \sum_{n=0}^{\infty}\left(\frac{n^{2}}{5^{n}}\right) \) :
\( \su
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(1 point) In this problem you will compute the value of ∑ n=0
[infinity]
( 5 n
n 2
). (a). Express 1/(1−x) as a geometric series: 1−x
1
=∑ n=0
[infinity]
(b). Differentiate both sides of the equation in part (a) with respect to x, expressing the right side as ∑ n=0
[infinity]
c n
x n
for constants c n
: =∑ n=0
[infinity]
(c). Multiply both sides of the equation in part (b) by x : =∑ n=0
[infinity]
(d). Differentiate both sides of the equation in part (c) with respect to x : =∑ n=0
[infinity]
(e). Multiply both sides of the equation in part (d) by x : =∑ n=0
[infinity]
(f). Reindex the right-hand side of the equation in part (e) to obtain a sum starting at n=1 (Left-hand side of equation in part (e))=∑ n=1
[infinity]
(g). Use your equation in part (f) to compute the value of ∑ n=1
[infinity]
( 5 n
n 2
) : ∑ n=1
[infinity]
( 5 n
n 2
)= (Check that the sum converges; if it does not, enter "diverges".) (h). Use your equation in part (g) to compute the value of ∑ n=0
[infinity]
( 5 n
n 2
) : (h). Use your equation in part (g) to compute the value of ∑ n=0
[infinity]
( 5 n
n 2
) : ∑ n=0
[infinity]
( 5 n
n 2
)= (Check that the sum converges; if it does not, enter "diverges".