- #1

keithk

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**Sum of a Series - 1983 BC 5 part C**

I don't have the actual problem (this is for a friend) but this is what I could gather from what she was saying.

## Homework Statement

If [tex]f(x) = \sum^{\infty}_{n=0}a_{n}x^n[/tex] find the value of f'(1)

[tex]a_{0} = 1[/tex] and [tex]a_{n} = (7/n)a_{n-1}[/tex]

## Homework Equations

None maybe?

## The Attempt at a Solution

Ok so after differentiating,

[tex]f(x) = \sum^{\infty}_{n=1}n a_{n}x^{n-1}[/tex]

Writing out the terms and subbing 1 for x got me to,

[tex]f(x) = \sum^{\infty}_{n=1}7^n/(n-1)![/tex]

or

[tex]f(x) = \sum^{\infty}_{n=1}n 7^n/(n)![/tex]

This was as far as I was able to get. Mathematica tells me that the answer is 7e^7.

I know that [tex]\sum^{\infty}_{n=1}n/(n)! = e[/tex] and that [tex] \sum^{\infty}_{n=0}7^n/(n)! = e^7[/tex] (which doesn't really help because we're

starting at 1). But with both parts in there I'm not sure what to do

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