# alternating

1. ### summation of alternating squared numbers

Hello I have been stuck on this exercise for quite a while now: Show that: \sum_{k=1}^{n}(-1)^k\cdot k^2=\frac{n\cdot(n+1)}{2} by using these two summation formulas: \sum_{k=1}^{n}(2k)^2=\frac{2n\cdot(2n+1)\cdot(n+1)}{3} and \sum_{k=1}^{n}(2k-1)^2=\frac{4n^3-n}{3} My initial guess was...
2. ### alternating test

why the author said the series 9 converges by using alternating test ? for the alternating test , the first term the sequence must be in descending order , the it is said to be converges , right? however , in series 9 , 1 is >(-1/2) , but (-1/2) is not > (1/3)
3. ### Limit: Sum to infinity of reciprocals of factorials with alternating sign

Hello, I'm a bit stuck on this and would value your help please. I know that the sum to infinity of the reciprocals of factorials is e. But what if they have alternating sign, e.g. 1/1! - 1/2! + 1/3! + 1/4! - ... I have found the answer: identity (9) at...
4. ### alternating series: this one i am clueless about

approximate the sum of the series correct to four decimal places. i have no idea how to do this one. $\sum_{n=1}^{\infty}\frac{(-1)^n}{3^nn!}$
5. ### Using Maclaurin Series to find the value of the alternating series

hey guys, I'm having problem with part C. of this question. I try to use geometric series to find the value but to no success. Can anyone tell me how should I deal with it? Junks
6. ### Can you use the Alternating Series Test on series that changes signs every 2 terms?

The Alternating Series Test (AST) says than an alternating series \sum (-1)^n u_n = u_1 - u_2 + u_3 - u_4 + ... converges if the u_n[\tex]'s are all positive, decreasing and its limit is 0. Does that exclude using it on terms that alternate every two or three terms? Even if it switches signs...
7. ### sum of alternating series

Hi, I am trying to prone that the sum of the following series is log3. 1+1/2-2/3+1/4+1/5-2/6+1/7+1/8-2/9+1/10+1/11-2/12... Thank's in advance.
8. ### Alternating Series - Converge or Diverge?

\sum_{n = 1}^{\infty} \dfrac{(-1)^{n}4}{n} a_{n + 1} = \dfrac{4}{n + 1} a_{n} = \dfrac{4}{n} n = 5 and a_{n + 1} = \dfrac{4}{6} and a_{n} = \dfrac{4}{5} How did they come up with the value of n? :confused: Alternating Series Test If 0 < a_{n + 1} \leq a_{n} and \lim n \rightarrow \infty...
9. ### Alternating groups

Hey guys. I'm sitting with an exercise I can't quite finish. Let \sigma \in S_6 be defined as follows: \sigma = (\frac{ 1 2 3 4 5 6}{4 5 2 1 6 3}) (of course it's not actually a fraction, I just dont know how to write a permutation.) I have already answered the first three exercises: (a) -...
10. ### Alternating Series - Convergence?

Determine whether the series \sum\limits_{n=2}^\infty (-1)^n \int_n^{n+1} \frac{dx}{ln^2 x} is absolutely convergent, conditionally convergent, or divergent. I cannot remember how to integrate \frac{1}{ln^2 x}. Should I try the comparison or limit tests?
11. ### alternating series

Hi, Can somebody please tell me how to find the sum of the following alternating series? \sum_{n=0}^{\infty} (-1)^n exp(-5*n)/(2*n+1) Thanks,
12. ### Alternating Group A4 and Cayley diagraph

Show that A4 = {(12)(34), (123)}, and draw a picture of the Cayley graph with respect to this generating set. I know that the elements generated by (12)(34) and by (123) are all in A4, but how can I show that there are 12 distinct elements, therefore showing that A4 is generated by those two...
13. ### How Do I Use The Alternating Series Test To Prove...

How would I use the Alternating Series Test to prove that sum from n=1 to infinity of ((-1)^n)/(ln(7n)) is conditionally convergent. Is there another test I have to use?
14. ### Partial Sum Approximation for Alternating Harmonic Series Question

1. The problem statement, all variables and given/known data Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. 2. Relevant equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . ...
15. ### Alternating Series Test

* I hope its alright that I reposted this. I realized that Calculus might not be the best category for this. * Show by example that the hypothesis b1>=b2>=...bn>=0 cannot be replaced by bk>=0 and limit k-->infinity =0 hint: use |ab|< 1/2(a2+b2) I've found an example: bk=(1/k2 + 1/k)...
16. ### Alternating Series Test

Show by example that the hypothesis b1>=b2>=...bn>=0 cannot be replaced by bk>=0 and limit k-->infinity =0 hint: use |ab|< 1/2(a2+b2) I've found an example: bk=(1/k2 + 1/k) which satisfies the limit going to zer and all terms being positive, which diverges. I'm getting stuck with a rigorous...
17. ### finite alternating harmonic series

This is similar to an existing thread, but sufficiently different, I think. For any positive integer n, let S(n)=\Sigma^n_{k=1}\left({1\over2k-1}-{1\over2k}\right)={u_n\over d_n} with u_n and d_n relatively prime. The question concerns u_n and d_n. One result. The exact power of 2 that...
18. ### alternating group

Hello! Can somebody help me with the following problem? Let A_5 be the alternating group of order 5. a) Which orders can a given element of A_5 possibly have? b) Let n be a possible order as in a). How many elements of order n does A_5 have? This seems to work with the Sylow theorems, but how...
19. ### Topology. Elementary alternating tensors.

Let x,y,z,w \in \mathbb{R}^5 Let T(x,y,z) = 2x_2 y_2 z_1 +x_1 y_5 z_4 S(x,y) = x_1 y_3 + x_3 y_1 R(w) = w_1-2w_3 (a) Write Alt(T) and Alt(S) in terms of elementary alternating tensors. (I think this is analogically to writing T and S in terms of alternating tensors ) (b) Express...
20. ### alternating series sum

calculate the sum of (-1)^n (n/8^n) i know fourier coefficients and bessels inequality how to evaluate this