If f(x)=∑(n=1 to ∞) (-1)^n*(x-π)^n/n^2 find f '''(x)
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# If f(x)=∑(n=1 to ∞) (-1)^n*(x-π)^n/n^2 find f '''(x)

[From: ] [author: ] [Date: 11-05-25] [Hit: ]
.This doesnt match the series that I get from the above sum, any ideas where Im screwing up?now it matches with term by term differentiation-To your first question, yes and no. You start moving up by 1 after you take the first derivative because you see,......
So I'm trying to find the 3rd derivative of f(x), these are the steps I've taken trying to differentiate it in the summation form:

f '(x)=∑(n=2 to ∞) (-1)^n*(x-π)^(n-1)/n

f ''(x)=∑(n=3 to ∞) (-1)^n*(n-1)*(x-π)^(n-2)/n

f '''(x)=∑(n=4 to ∞) (-1)^n*(n-1)*(n-2)*(x-π)^(n-3)/n

My first question is do I move up the counter by 1 every time I take the derivative? Also do I have to keep incrementing it up for the (-1)^n as well?

Secondly, when I do term by term differentiation I get the following series for the 3rd derivative:

{-2/3 + 3(x-π)/2 - 12(x-π)^2/5 + 10(x-π)^3/3 +...}

This doesn't match the series that I get from the above sum, any ideas where I'm screwing up? Thanks

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you need not move up the counter for first derivative

It is actually

f '(x)=∑(n=1 to ∞) (-1)^n*(x-π)^(n-1)/n

f ''(x)=∑(n=2 to ∞) (-1)^n*(n-1)*(x-π)^(n-2)/n

f '''(x)=∑(n=3 to ∞) (-1)^n*(n-1)*(n-2)*(x-π)^(n-3)/n

now it matches with term by term differentiation

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To your first question, yes and no. You start moving up by 1 after you take the first derivative because you see, the term n=1 for f'(x) still exists, it's a scalar. The reason you move it up by 1 in the seceding derivatives is because this (and those of the following derivatives) scalar becomes 0. So n=2 for f''(x) and n=3 for f'''(x)

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You don't move the counter unless the first term is 0.

f '(x)=∑(n=1 to ∞) (-1)^n (x - π)^(n-1)/n, the index still starts at 1, since the first term is nonzero.

f ''(x)=∑(n=2 to ∞) (-1)^n (n-1)*(x - π)^(n-2)/n, since n-1 = 0 for n = 1.

f '''(x)=∑(n=3 to ∞) (-1)^n (n-1)(n-2) (x - π)^(n-3)/n, since n-2 = 0 for n = 2.
.......= -2/3 + 3(x - π)/2 - 12(x - π)^2/5 + ..., as required.

I hope this helps!
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