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Taylor series (centered at -1) is given by:

$$ \sum_{n=1}^\infty \frac{(n+1)}{n}(x+1)^n $$

  1. what function centered at -1 does this series represent?

  2. hints as to how I may find its interval of convergence is (-2,0)?

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    Thanks for editing the question, with appropriate symbols Matt. My first time here actually.2012-09-06

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Hint: Let $w=1+x$. Note that $\dfrac{n+1}{n}=1+\dfrac{1}{n}$.

So our sum is $$\sum_1^\infty w^n +\sum_1^\infty \frac{1}{n}w^n.$$ The first sum will be very familiar. For the second, note that $\dfrac{w^n}{n}$ is an antiderivative of $w^{n-1}$.

For convergence, you are interested in showing that the interval is $-1\lt w\lt 1$. Ratio test will do it, except that you need to show also that we do not have convergence at $w=\pm 1$.

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    Left term in sum: {1/(1-w)} - 12012-09-06
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    @student101: Indeed, and now you should be ready for right term.2012-09-06
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    right term in sum: {1/(1-w)}d/dw= 1/{(1-w)^2}2012-09-06
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    @student101: Integrate, don't differentiate. Answer looks like $\int_0^w f(t)\,dt$ for suitable $f$.2012-09-06
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    integral being -log(1-w)2012-09-06
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    @student101: $f(t)=\frac{1}{1-t}$, the integral is a logarithm. Need to smoke, will be away from computer for a while.2012-09-06
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For the interval of convergence, consider the root test, and answer the following questions:

What is $\lim_{n \to \infty} (\frac{n+1}{n})^{1/n}$?

For what values of $x$ is $|x+1| < 1$?

For what values of $x$ is $|x+1| = 1$?

When $|x+1| = 1$, does the series converge or diverge?

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    so (1+1/n)^(1/n) converges to 12012-09-06
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    abs(x+1)<1 for -22012-09-06
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    at x=0, the series converges; thanks :)2012-09-06
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    at x=-2, series converges also2012-09-06
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    How can it converge at $|x+1| = 1$ when the absolute value of each tern is greater than $1$?2012-09-07
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    not sure if i'm misunderstanding but here is how i see it: abs(x+1)=1, implies x=0. Now at x=0, the series in original post reduces to SUM({n+1}/{n}); which converges to 1 as n approaches infinite.2012-09-08