[Math] Given the taylor series, find the function it derives from

calculussequences-and-seriestaylor expansion

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)?

Best Answer

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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