Infinite series Definition and 390 Threads

Homework Statement I am supposed to find the value of the infinite series: \sum_{n=0}^{+\infty}\frac{\pi\cos(n)}{5^n} Homework Equations I asked this question before on this forum and micromass told me that I should use cos(n)=((e^i)^n+(e^(-i))^n)/2. That equation worked and I was...

Homework Statement (Sorry, but I haven't mastered using the sigma notation in these forums yet). Find the sum of the following infinite series: (n=0)^(inf) SIGMA ((pi)cos(n))/(5^n). Homework Equations I tried using the formula S=(a1)/(1-r). I know that a=pi, but I can't find "r."...

I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me. Suppose we have an infinite series of the form: \sum^_{n = 1}^{\infty} 1/n^\phi where \phi is some even natural number, it appears that it is always...

Homework Statement [PLAIN]http://img818.imageshack.us/img818/817/potentialenergy.jpg Homework Equations Given above. The Attempt at a Solution I've never had physics in my life and am completely baffled by this problem. I'm only in calculus 3 and am just learning infinite...

Homework Statement 2/3!+4/5!+6/7!+...to infinity is equal to?

I think I did this right... \sum_{i=1}^{\infty} \frac{n}{e^{n^2}} I tried it with the root test to no avail. So I then tried it with the Ratio Test and I come to this expression... \lim_{n \to \infty} \frac{(n+1)}{e^{2n} e(n)} ...which is an indeterminate form (infinity over infinity)...

Homework Statement Suppose that {an} is a monotone decreasing sequence of positive numbers. Show that if the series an converges, then the lim(nan)=0. Homework Equations The Attempt at a Solution I started the proof with the fact that since I know the sequence is monotone decreasing and the...

Homework Statement http://img840.imageshack.us/img840/3609/unleddn.png note that by log(n), i really mean NATURAL log of n Homework Equations it's convergent, but I can't figure out which test to useThe Attempt at a Solution there is no term to the nth power, so ratio test is useless; root...

I would like to simplify this series as much as possible f(m)=\sum_{n=0}^{\infty}\frac{m^n (2n)!}{(n!)^3} Approximates would also be fine. One can easily notice that (2n!) / (n!)^2 > 2^n hence I figured out that f(m) > \sum_{n=0}^{\infty}\frac{(2m)^n}{n!}=\exp(2m) but this is not the best...

Hello, Could anybody help with this series: $\sum_{n=0}^\infty e^n/(e^n+1)^{a-1},\,\, a>2. $ I tried (without success) to adapt the Riemann integral theorem and the laplace transform. For the latest, I will need to find the inverse laplace transform of $e^n/(e^n+1)^(a-1)$, which does...

Homework Statement \int_0^x \frac{1-cos(t)}{t} Homework Equations The Attempt at a Solution I'm lost completely. If I separate it and then try integrating it has 0 for the ln(x) which has to be wrong.

Basically, find the limit of the sequence: {[(n+3)/(n+1)]^n}, from n=1 to infinity Book says it's supposed to be e^2, and indeed the graph shows that... I'm not sure what to do with the top of the fraction. Working with the bottom and dividing by n, I obtain, lim as n approaches...

Homework Statement I have been straining to find convergence or divergence of a few infinite series. I have tried everything I can think of; ratio test, root test, trying to find a good series for comparison, etc. Here are the formulas for the terms: #1 1 ------------- (ln(n))^ln(n)...

Homework Statement ∞ ∑ ( 60^(1/(n+3)) − 60^(1/(n+4)) ) n = 0 Homework Equations I believe this is a geometric series so the sum would equal a/(1-r) The Attempt at a Solution I tried to view it as a geometric series but i had trouble finding a ratio, especially what i thought...

Is it possible and is there a general method?

Homework Statement show \sum 1/(ln k)^n diverges,for any n. the indexing is k = 2,3,... Homework Equations The Attempt at a Solution Because k > ln k, k^n > ln k^n, and 1/k^n < 1/ln k^n so this is just a p-series, which diverges for p =< 1. So now I need to show it diverges for n > 1...

I have to show that ln(1 - 1/k^2)= - ln(2) I took LCM and separated the equation using ln(A/B) = lnA - lnB How should i proceed now?

Homework Statement I have an assignment to write a program for calculating the sine (and various other functions) using the method of truncated infinite series using DO statements. The DO statement is supposed to run until the difference between the current and last iterations are less than...

Prove that sum 1/2^(n+1)*n/(n+1), from n=0 to infinity, converges to 1 - log(2), where log stands for the natural logarithm. I know that the Taylor series for log(x) about x=1 is sum (-1)^(n+1)*(x-1)^n/n, but I don't see how these two statements are consistent. Thanks for any pointers!

\sum [(-1)^n * pi ^2n] / [9^n * (2n)!] = ? Thanks for the help.

Homework Statement Show the convergence of the series \sum_{n=1}^{inf}(\frac{1}{n}-\frac{1}{n+x}) of real-valued functions on R - \{-1, -2, -3, ...\} .Homework Equations The Attempt at a Solution I first thought of solving this using telescoping series concept but it didn't work out. Also, I...

Homework Statement Find the sum of the series: (1/(4^n))+ (((-1)^n)/(3^n)) from n=0 to infinity Homework Equations The Attempt at a Solution I'm not overly familiar with series and am not sure how to approach this. A lot of help guides online talk about testing for...

Homework Statement Prove the convergence or divergence of the series \Sigma(\frac{n}{2n+3})^{2} using the Direct Comparison Test. Homework Equations If series A converges and every term in series B is less than the corresponding term in series A, then series B converges. If series C...

Hi all,SUM of Series from n=2 to infinity of: 1 ------------ (2^n) (n-1) This question is driving me bananas... my tools are Telescoping or Geometric series, but neither seem to work: I've tried everything to get this into a geometric series form and then using the a/1-r formula, but...

Homework Statement Sum from 0 to infinity of (2^n + 6^n)/(2^n6^n) Homework Equations No idea. The Attempt at a Solution I am completely dumbstruck on how to do this one. Could someone give me a hint on where to start? Thanks a lot!

\sum\frac{1}{n(n+k)} from n=1 to infinity find that the series is convergent and find it's sum. Now I'm a bit confused... I can show it's convergent with k=1 and I attempted the same thing with k by breaking this into partial fractions. But I'm given a harmonic series that is divergent...

Homework Statement What is the value of: 1 + (\frac{1}{3})^{2} + (\frac{1}{5})^{2} + (\frac{1}{7})^{2} + (\frac{1}{9})^{2} + ...

Is there a physical intrepretation of infinite series? Is there a picture that will explain what is meant by infinite series or is there a tangible application of this concept.

Finding "a" and "b" in an infinite series limit comparison test Homework Statement \sum_{n=1}^\infty \frac{\sqrt{n+2}}{2n^2+n+1} How do I identify my a_n and my b_n? In this particular problem you need to use the Limit comparison test which is your "a_n" divided by your "b_n". I...

Homework Statement \sum_{k=2}^\infty \frac{k^2}{k^2-1} \sum_{n=1}^\infty \frac{1+2^n}{3^n} Homework Equations I know that for the first problem i can apply the Divergence test by finding my limit as K goes to infinity. By doing this i get 1 which does not equal zero so i...

Homework Statement suppose a large number of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction of particles reflected, so then you can assume (1-r) is the number of particles that escape at each wall. Suppose particles...

I looked through some tutorials to find intervals of divergence and tests for divergence... My series: [PLAIN]http://img843.imageshack.us/img843/4193/51453212.jpg a and x are constants... I did the ratio test and i get \rho=1, so i tried to apply the limit test to see if an is zero or does...

Homework Statement In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n=2, the water can be made a pure as you like, but if n=3, at...

Homework Statement I want to find a closed form formula for: x+2x^2+3x^3+4x^4+\ldots I know that this can be written as: \sum_{n=1}^{\infty}nx^n but I would like to have a closed formula for this. The formula for an infinite geometric series is: \sum_{n=0}^{\infty}x^n =...

Homework Statement Test if the infinite series converge or diverge. Homework Equations \sum_{n=1}^{\infty}\frac{4n+3}{n(n+1)(n+2)} The Attempt at a Solution I tried Ratio test: a_{n+1} = \frac{4n+7}{(n+1)(n+2)(n+3)} a_{n} = \frac{4n+3}{n(n+1)(n+2)} \left|\frac{a_{n+1}}{a_{n}}\right| =...

I am having a difficult time working with some of these infinite series. I studied them in calc 2, but that was a few years ago. Could someone help me figure out how to find what the following sum converges to: \sum_{n=-N}^N{cos(n \theta)} Shouldn't there be some property...

I know that \sum_{n=-\infty}^\infty{1} = \infty But I don't understand why. It seems to me that since the constant inside the summation is not dependent upon n it can be moved outside the summation. Then there is nothing to sum. It seems to me that...

I understand how this works: \cos x = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \ldots But what about this? \frac{1}{\cos x} = \frac{1}{0!} + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} +...

Homework Statement Given two convergent infinite series such that \sum a_n -> L and \sum b_n -> M, determine if the product a_n*b_n converges to L*M. Homework Equations The Attempt at a Solution If know that if a_n -> L this means that the sequence of partial sums of a_n = s_n...

Hi all, I've been messing around with the product of Poisson distributions and was hoping someone could help me work out a closed form solution for the following convergent infinite series (given x > 0): \sum^{\infty}_{i=0} \frac{x^{i}}{i! \times i!} Many thanks in advance, Jacob.

Supose a_n=f(n) What is the relationship between convergence or divergence of: \sum_{n=1}^\infty a_n and \int_{1}^{\infty}f(x)dx ? Besides the integral test (which only works for special cases).

Homework Statement Given an infinite series that follows the form [(xlna)^(n-1)]/n! n takes on integers from 0 onwards x all real numbers a all positive real numbers Homework Equations Maclaurin series expansion The Attempt at a Solution In which for the e^x series expansion...

Infinite Series!??COnverge or Diverge Homework Statement 1. ∑(infinity, k=1) 5k^(-3/2) 2. ∑(infinity, k=1) 1/(k+3) Homework Equations converge or diverge The Attempt at a Solution 1. converges p=3/2... 5/(infin)=> 0 2. diverges p=1 I still don't get why 2 diverges? 1 converges...

how would i go about finding the sum of the infinite series 2^k/k!? its not a geometric so i can't use the formulas for that so i really have no clue. any help would be appreciated

Homework Statement http://bit.ly/9N9iLZ Evaluate: lim n-> infinity of Sum (from k = 1 to n) of sqrt(k/n) * 1/n Homework Equations taylor series? The Attempt at a Solution the above = lim n->infinity of Sum (from k = 1 to n) of k^1/2 / n^3/2 k approaches n so n^1/2 / n^3/2 ->...

Homework Statement I don't have the problem in front of me but it was something like "converge or diverge"?: \sum 5^n/(n+1) The Attempt at a Solution I would like to know that if I use the root test, would I get lim n-> \infty 5/(n+1)^1/n = 5/(n+1)^0 = 5/1 = 5 ? I suspect...

Homework Statement Rewrite the given expression as a sum whose generic term involves xn [m=2 to ∞] ∑m(m-1)amxm-2 + [k=1 to ∞] x∑kakxk-1 Homework Equations None in this problem The Attempt at a Solution To make the first part involve only xn, I can use the substitution n=m-2...

Homework Statement I'm told to evaluate the following to the thousandths place: \infty \Sigma 7*(0.35)^k k=1 Homework Equations We know that an infinite equation can be expressed as: S\infty=(a1)/1-rn The Attempt at a Solution The first term (a1) is 7 and r=.35 so I can...

Homework Statement Find the sum of the following series. SUM (n=1 to inf) -3^(n-1)/(8^n) Homework Equations Possibly fit into ar^n format? [b]3. The Attempt at a Solution [/b I feel there is a way that this fits into a geometric form in which case could use a/(1-r) to find...

Infinite series sin(1/n)/n ? Homework Statement does the series (sin (1/n)) / sqrt ( n ) converge or diverge? (series from n = 1 to infinity...) Homework Equations The Attempt at a Solution I thought that for this we could do a comparison of sin (1/n) to a finite number...