pineapplechemist
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We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
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Gome44
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(Original post by pineapplechemist)
We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
For i, consider the expansion of e^x
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ghostwalker
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(Original post by pineapplechemist)
...
Presumably you're already familiar with the power series for some functions.

The first is not that dissimiliar to a well known series - can you adapt it?

The second, if you differentiate it, is a standard series. So, do that, work out the function, and integrate the function to get the function for your original series.
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Gome44
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For ii) it is a standard maclaurin series after differentiation
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ztibor
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(Original post by pineapplechemist)
We need to find a simple expression for the following infinite sums:

(i) 1-x+(x^2)/2! - (x^3)/3! +....
(ii) (x^2)/2 - (x^3)/3x2 + (x^4)/4x3 - (x^5)/5x4 +... for modulus of x<1

The hint for the second one is to differentiate. Still not sure how to go about these. We need to write them as functions apparently, not using sum notation.
for ii) :

 \displaystyle \frac{x^n}{n \cdot \left(n-1\right )}=\frac{x^n}{n-1}-\frac{x^n}{n}

Grouping

\displaystyle \left (\frac{x^2}{1}+\frac{x^3}{2}+ \frac{x^4}{3} + ...\right )-
\displaystyle -\left (\frac{x^2}{2}+\frac{x^3}{3}+ \frac{x^4}{4} + ....\right )

i.e the second group with pointwise differentiation

\displaystyle -\int \left (x+x^2+x^3+ ...\right ) dx =-\int \frac{x}{1-x} dx

which you can integrate easily

Similarly for the first group but first take out x as factor before the differentiation
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