Alternating series Definition and 109 Threads

am trying to find the intervals of convergence for the summation, first deritive, and intergral of problems like this: the sum of [(-1)^n+1(x-5)^n]/[n5^n] from n=1 to infinity i know it is an alternating series and thus i am attempting to use that test to find convergence/diverigence lim...

I apologize right now for the fact that I have no idea how to use LaTeX I can't figure out if the following alternating series is convergent or not: Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so I have...

I apologize right now for the fact that I have no idea how to use LaTeX I can't figure out if the following alternating series is convergent or not: Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so...

Alternating Series Help! I apologize right now for the fact that I have no idea how to use LaTeX I can't figure out if the following alternating series is convergent or not: Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity the root test is not applicable, A(n+1)>An, and the ratio...

\sum_{n=1}^{\infty} a_n = 1 - \frac {(0.3)^2}{2!} + \frac {(0.3)^4}{4!} - \frac {(0.3)^6}{6!} + \frac {(0.3)^8}{8!} - ... how many terms do you have to go for your approximation (your partial sum) to be within 0.0000001 from the convergent value of that series? the answer to this...

I know that a series such as \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} is divergent. Is this also the case for an alternating version of the same series, i.e., \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}} ?

We were given the series: 1/2^6 - 1/2^8 + 1/2^10 - 1/2^12 + ... And asked to find the general term, an, which I worked out to be (-1)^(n+1)/2^(4+2*n). To see if it converged, I used the alternating series test and found that it converged ie the lim as n tends to infinity = 0 and the...

Consider the following: \int _0 ^1 \sqrt{1+x^4} \mbox{ } dx = \left[ x + \frac{x^5}{2\cdot 5} - \frac{1}{2!2^2 9}x^8 + \frac{1\cdot 3}{3!2^3 13}x^{12} - \frac{1\cdot 3\cdot 5}{4!2^4 17}x^{16} +\dotsb \right] _0 ^1 According to the alternating series estimation theorem, we find...

Again, my rusty algebra and derivative taking is getting me into trouble. This is from the section on alternating series. Overall, I think I'm getting the concepts, but some of the solutions to the problems are leaving me scratching my head. \sum \frac{(-1)^{n+1}(n+1)}{ln(n+1)} How did...