Taylor series (centered at -1) is given by:
$ \sum_{n=1}^\infty \frac{(n+1)}{n}(x+1)^n $
what function centered at -1 does this series represent?
hints as to how I may find its interval of convergence is (-2,0)?
Taylor series (centered at -1) is given by:
$ \sum_{n=1}^\infty \frac{(n+1)}{n}(x+1)^n $
what function centered at -1 does this series represent?
hints as to how I may find its interval of convergence is (-2,0)?
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 $w=\pm 1$.
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?