Infinite series Definition and 390 Threads

1. Find whether the following series converges or diverges or is oscillatory \sum1/(n^3*(sin^2 n))

Homework Statement Find the sum of the series: ln(n)/n^2 from n=1 to infinity. I already know that it is convergent(at least i hope i am right on that fact) Homework Equations The Attempt at a Solution I tried to use geometric series but i can't see anything like that that would...

1. Problem Prove that \sum_{n=0}^{\infty} \frac{a}{k^n} = a\frac{k}{k-1} Homework Equations - The Attempt at a Solution Don't know how to do it at all.

1. Examine the series \frac{1}{1 . 2} +\frac{1}{2 . 3}+\frac{1}{3 . 4}+\frac{1}{4 . 5}... for convergence. 3. The Attempt at a Solution The following is the book's answer: "lim_{n\rightarrow \infty}S_{n} lim_{n\rightarrow \infty} (1 - \frac{1}{n + 1}) = 1 - 0 = 1 Hence the...

Concept question: What are they used for? I understand functions used for position/time/velocity etc., but what are infinite series actually used for? Are they just a sum of numbers with no application? I'd like to know what I'm devoting my brainpower to before I spend massive amounts of...

Homework Statement a) consider the infinite series (k=1) sum (inf) [(k+1)^(1/2) - (k)^(1/2)] expand and simplify the nth partial sum. determine wether the oartial sums S_n converge as n-> inf b) determine all the numbers x in R so that the infinite series (k=0) sum (inf) [x^(k)/(k!)]...

Hello, folks. This happens to be my first post here, and I've come with a question from a problem set in my textbook. Homework Statement Determine whether the following series converges or diverges. Give reasons for your answer. Homework Equations \sum^{\infty}_{n=2}...

Homework Statement Find the sum of the infinite series: 405-270+180-120+80... Homework Equations ?? The Attempt at a Solution I know there's a formula for this but I can't remember it. Could someone refresh my memory?

Homework Statement How can I compute ∞ ∑ (1 / y!) ? y=0 Homework Equations N/A The Attempt at a Solution In the middle of a problem from a statistics course, I got this series and forgot how to evaluate an infinite series in general and in particular this one...Please help!

Homework Statement Evaluate ∞ ∑ [(e-15 15x) / x!] x=16 15 ∑ [(e-15 15x) / x!] x=0 Homework Equations The Attempt at a Solution The only way I can think of is writing out every term explicitly and adding them on a calculator. Is there any faster way (without having to...

Homework Statement [sum of; n=1; to infinity] ((2y)^n)/(n(n!)) Homework Equations The Attempt at a Solution If there were a way to find the improper integral of f(x) = ((2c)^x)/(x(x!)) from one to infinity using unit step integration (c is just a constant), then that would equal...

Please I have a similar problem, how can I compute the sum of this infinite series: SUM(X / (Y^X) ); where X(i)>0

How do you go about finding the sum, \sum \frac{1}{n^2}. I remember studying it earlier, but don't quite remember how it was done..just tell me the method. i'll figure the rest out.

infinite series solution for NON-linear ODEs? Is it possible to use the infinite series method (Frobenius) to obtain general solutions of non-linear ODE's, I want to try a second order equation. Any good references where I can see how that goes exactly?

Homework Statement I am to find the sum of the series, but what do i do if it is infinite?? no clue. i'm also not sure how to type the symbols so i hope you can understand me:shy: : (Sum) n=0, limit = infinity: 1/2(2/3)^n Homework Equations i 'm not sure. The Attempt at a Solution...

I am getting into this topic and I am having a hard time conceptualizing it. Is there anybody that can spend a minute letting me know the "reality" to infinite series? By that I mean, please explain infinite series in such a way that a beginner like me will be able to take what you said and...

My Calculus teacher posed this question to a recent class, and asked us (previous students) if we could figure it out(just for fun). I am stumped. The question is to find the general formula that represents the infinite series (1, -1, -1, 1,-1, -1, 1...) I am assuming it uses trig graphs...

hello. I have transformed the Laplace transform into the infinite series by repeatedly using integration by parts. What is this infinite series? may be Laplace transform series, or only an infinite series without name? L(t)= \int_{t}^{\infty}\frac{f(t)}{e^{st}} dt =-0 +...

Homework Statement Find if the series is absolutely convergent, conditionally convergent, or divergent. The sum from two to infinity of (-1)^n/lnx. Homework Equations The Attempt at a Solution I don't know how to integrate 1/lnx, so that failed. The ratio and root test don't...

Homework Statement Show, from the definition of continuity, that the power series function f(x)=sum(a_n*x^n) is continuous for its radius of convergence.Homework Equations Definition of continuityThe Attempt at a Solution Must show that for any |a| < R, given e>0 there exists d>0 such that...

(1/2) + (2/4) + ... + (n/(2^n)) = sum i=1 to i=infinity of (i/(2^i))?i know how to express the sum of just 1/(2^i), but not the above thanks for the help!

Homework Statement I need to show that \Sigma\frac{1}{nlnnlnlnn} from n=27 to n=10^(100,000) is approximately equal to 8.1

Homework Statement Estimate \sum^{\infty}_{n=1}n^{-3/2} to within 0.01 Homework Equations \int^{\infty}_{n+1}f(x)dx\leq R_{n} \leq \int^{\infty}_{n}f(x)dx The Attempt at a Solution So my strategy was using the above formula to find Rn, where Rn = 0.01 or 1/10^2. Then that will give me the...

Homework Statement The infinite Series starts at n=1 and is (4-sin(n))/(n^2 + 1) For each series which converges, give an approximation of its su, together with an error estimate, as follows. First calculate the sum s_5 of the first 5 terms, Then estimate the "tail" which is the infinite...

What exactly is the difference between "increasing/decreasing" and "strictly increasing/decreasing" ? is it similar to conditional and absolute convergence?

given the series g(x)= \sum_{n=0}^{\infty}\frac{a_{n}}{\sqrt {x-n}} where the coefficients a_n are real numbers my question is does the above makes sense ? i mean since we are summing over all positive integers , no matter how big we choose 'x' there will be a factor so x-n...

[SOLVED] Infinite Series Homework Statement ln((n!e^n)/n^(n+1/2)) Homework Equations Does the series above converge or diverge. The Attempt at a Solution I can see that it diverges but I'm looking for the appropriate test to show this

Homework Statement Well we are given a series of steps done with the number "x" and in the end the end value is ln(x). Basically we are asked to prove why it isn't a coincedience Homework Equations I put the steps into an equation, but i can't prove it. ln(x) =^{lim }_{n->inf} (x^\frac{1}...

Im not sure if it is related to calculus but, Calculate the sum \sum^{\infty}_{n=0}\frac{(n-1)(n+1)}{n!} exactly. I tried to to partial fraction decomposition but couldn't find anything.

Homework Statement The question is to evaluate the infinite series of the Sum[(((-1)^n)*a(n))/10^n], as n goes from zero to infinity, and a(n) is the recurrence relation a(n)=5a(n-1)-6a(n-2) where a(0)=0, and a(1)=1 Homework Equations I found the explicit equation for a(n)=3^n - 2^n...

I'm having trouble picking apart this summation: \sum^{inf}_{n=1} P(E)*P(1-p)^{n-1}; where p = P(E) + P(F) I know I need to use the identity of a geometrical series when |r| < 1 : 1/(1-r) I'm getting P(E)/(1-(P(E)+P(F)) But I need to be getting P(E)/((P(E)+P(F)); The entire...

I have 2 questions I am having problems with. The goal is to determine if the series converges or not. Q1: Sum from(1 to inf) of (exp^i)/( (exp^2i) + 9) I tried to do the integral test but I cannot seem to integrate. Any guides would be appreciated. Also if I wanted to compare...

So I am supposed to show that the Infinite series \sum^{\infty}_{k=1}\frac{3}{k+4} does not converge using any method. Now, my question: Is \frac{3}{k+4} the General term? I will wait for a response before I continue, for it may eliminate another question regarding the General Term and Closed...

Question: Test for convergence: \sum\frac{n!}{10^n} (the sum is from 1 to infinity) I tried using \frac{n^n}{10^n}\geq\frac{n!}{10^n}\geq\frac{n}{10^n} and showing that either the first one was convergent or the last one was divergent using various tests but didn't get anywhere. Any hints?

Homework Statement When dropped, an elastic ball bounces back up to a height three-quarters of that from which it fell. If the ball is dropped from a height 2 m and allowed to bounce up and down indefinitely, what is the total distance it travels before coming to rest? Homework...

[Solved] Infinite Series Homework Statement Find the sum of the given series, or show that the series diverges. _∞ ∑_(k=0) (2^(k+3))/(℮^(k-3)) I hope this is not confusing, and it would be great if someone knows about some site where you can write equations easily online...

Homework Statement I'm looking to find a closed form for the infinite series: 1*C(n,1) + 2*C(n,2) + 3*C(n,3) + ... + n*C(n,n) Homework Equations C(n,k) = n!/(k!*(n-k)!) C(n,1) + C(n,2) + C(n,3) + ... + C(n,n) = 2^n - 1 The Attempt at a Solution I'm not quite sure where to start...

can anyone find a solution without using a calculator?? This is the problem: Find the positive interger k for which \sum \limits_{n=4}^k {1 \over \sqrt{n} + \sqrt{n+1}} = 10

Infinite Series (2 diverge --> 1 converge) I've been trying to figure this question out: Find examples of two positive and decreasing series, \sum a_n and \sum b_n , both of which diverge, but for which \sum min(a_n,b_n) converges. It doesn't make any sense to me that any positive and...

[SOLVED] Infinite series help Homework Statement \sum- (\frac{5}{4})^n i=infinity and n=0 Homework Equations Convergence of a geometric series \sum (ar)^n = a/(1-r) when 0<|r|<1 The Attempt at a Solution I have to explain why this series diverges or converges. The test for divergence gives...

Homework Statement I am wondering if someone could give me some insight on how the following infinite series was derived: P_e = \sum_{-\infty}^\infty (1/2)^{2|n|} = -1 + 2 \sum_{n=0}^\infty (1/2)^{2n} = 5/3 Homework Equations See above The Attempt at a Solution I think the -1...

Homework Statement the repeating decimal . 27272727 . . . can be written as an infinite series. Write it as a series and tell if it diverges/converges. If it converges, find the sum. Mickey, also a former student, knows how to do this one. Mickey knows enough math to write it as .27...

Homework Statement 1+ \frac{\alpha\beta}{\gamma} x + \frac{\alpha (\alpha+1)\beta(\beta+1)}{1.2.\gamma(\gamma+1)}x^{2}+... Homework Equations The Attempt at a Solution Using D'Alembert's ratio test, I get lim_{n\rightarrow\infty}\frac{U_{n+1}}{U_{n}}=x so, x>1 diverging series...

1) Determine whether the infinite series ∞ Sigma (k^2-1) / (3k^4 + 1) k=0 converges or diverges. [My immediate thought was to use the "limit comparsion test", but this test requires all terms to be positive. However, the first term (put k=0) is definitely negative...what should I do? Can...

Homework Statement It is not a homework problem.I just want to clarify whether during the comparison tests of infinite series,should we start the series from n=0 whenever possible? Homework Equations The Attempt at a Solution Actually,there are many series for which n=0 term is not...

If I take the limit on the sum... I get 1/1 = 1 If the limit does NOT = 0 then sigma f(x) diverges...I'm not quite sure I follow this... Does this mean that in order for the equation to converge, the sum (sigma) must be = to 0?

i'm not quiet sure how to attack this problem: sigma (2^n)+1/(2^(n+1)) n->1 If I start plugging in #'s for n, then I get: n=1: 3/4 n=2: 5/8 n=3: 9/16... by this method, I see that it's going to 1/2, but I need another way to 'see' that. Any suggestions?

Homework Statement I don't know if there is an analytic expression of this infinite series: f(x,y)=\sum_{n=0}^{+\infty}\frac{x^n}{1-y^n} here x,y<1 Homework Equations This series is convergent, so maybe it can be expressed as some special function?The Attempt at a Solution I tried to...

I don't know how to get a analytic expression of this infinite series: \sum_{n=0}^{+\infty} \frac{1}{1+x^n} here x>1. Thanks!

Homework Statement IS=infinite series from n=1 to infinity Does this IS(((-1)^n)/5) converge? But if it did what would it converge to? There is certainly nothing infinite about this sum value but if it dosen't converge to any specific number than does it mean it is divergent? I...