how to find a function expression of a power series?

Asked Jan 25 '19 at 14:32

Active Jan 25 '19 at 14:40

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I have the following power series: $x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\cdots$.

I want to use the sum of the geometric series $\sum^\infty_\limits{k=0}{x^k}=\frac{1}{1-x}$ to write a function $f(x)$ for the power series above.

When I look at the series I see that its equal to $\sum^\infty_\limits{k=0}{\frac{x^{k+1}}{k+1}}$, which is the integral of the geometric power series.

$$\int{\sum^\infty_{k=0}{x^k}=\int\frac{1}{1-x}}\implies \sum^\infty_{k=0}{\frac{x^{k+1}}{k+1}}=-\ln|1-x|$$

The function expression should be $f(x)=-\ln|1-x|$.

Is it correct to do it this way? Am I missing constant values after integrating?

edited Jan 25 '19 at 14:40

KM101

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asked Jan 25 '19 at 14:32

user3133165

I'm trying to relate this to the geometric series – user3133165 Jan 25 '19 at 14:36
If you plug $0$ in the initial power series, you should get the same value with the analytical function. – Jan 25 '19 at 14:37
So is this correct? does f(x) equal to the series for all x? – user3133165 Jan 25 '19 at 14:39
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I haven't looked at your calculations, but yes, differentiating and antidifferentiating are common ways of trying to find closed forms for power seies. – Arthur Jan 25 '19 at 14:39
Is there a way to make sure that f(x) is the correct function for the series ? – user3133165 Jan 25 '19 at 14:40
If you write down the Taylor series of $f$, you should get your original series. – Klaus Jan 25 '19 at 14:41
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"Does $f(x)$ equal to the series for all $x$?" No. The series does not converge everywhere, only for $|x|<1$. – kccu Jan 25 '19 at 14:44
So am i to understand now that for $|x|\le1$ both f(x) and the series are converge and are equal? – user3133165 Jan 25 '19 at 14:50
Can't use $x=1$ otherwise get $-\ln(0)$. – coffeemath Jan 25 '19 at 16:30
Finally, $ f(x)=-\ln(1-x)$ equals the series for $-1 \le x < 1$. For other values of $x$, the series diverges. – Andreas Jan 27 '19 at 20:38

how to find a function expression of a power series?

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