Infinite series Definition and 390 Threads

Homework Statement Test to see whether the following series converges \sum_{n=1}^\infty \sqrt[n]{2}-1 Homework Equations All we've done so far is integral test, ratio test, and root tests.The Attempt at a Solution As n approaches infinity, the term apporaches 0, so it may or may not...

What Is "Limits and Infinite Series"? I signed up for my classes for spring quarter and I had room for another math class. Since I'm dual majoring in math and physics I figured I should take as much math as I could. I had to sign up for Multivariable Calculus I so I could take Electricity and...

i have to prove that the sum of the follwing infinite series is irrational. \frac{1}{2^3} + \frac{1}{2^9} + \frac{1}{2^{27}} + \frac{1}{2^{81}} + \cdots i have no idea where to begin. thanks in advance. note: this is not a homewrok problem.

Homework Statement Find the sum of the infinite series, if the series converges. infinity E n = 1 (2) / (n^2 + 2n) Homework Equations .. The Attempt at a Solution I believe this problem doesn't look very hard, I think all i really need to do is divide the...

How do I evaluate the infinite series (2^n)/(n!) from n=0 to infinity? (I don't know how you put the little E symbol and all that in so I had to write it out.) I already found that it converges to 0 using the ratio test, but I don't know quite how to evaluate it. Any help would be mondo...

Hey guys! My friend showed me a infinite series on maple infinity --- \ ( ((n-1)^2) - (1/4) )^(-1) = (Pi/2) / --- n=1 Is anyone familar with this? If so can you refer me to its proof? Or maybe post it. Thanks

I need help with the following problems: 1. Prove whether: sum from x=1 to infinity of x!*10^x/x^x converges or diverges 2. Prove whether: sum from x=3 to infinity of sqrt(m+4)/(m^2-2m) converges or diverges 3. Calculate the Maclaurin series of f(x)=3x^2*cos(x^3) Hint: Explicity...

Since the forum seemed to chew up my post last time, I thought I'd give it another try... I'm looking to evaluate a series of the form \sum_{n=0}^\infty \frac{1}{(n-f)^2} where f is a constant between zero and one. I really have no clue where to start on this. I've seen series such as...

Hello i have the infinite series 7^(K+1)/2^(3k-1) How do i find what it converges to if it does converge. Limit comparison does me no good. I am thinking integral and ratio test. root test does me no good either.

Show that Σ_{n=1}^\infty ln(1 -1/n^2) = -ln 2 I'm not sure how to do this. Should I use telescopic sums or should I make a function y = ln(1 - 1/n^2)? Is it possible to use telescopic sums here?

I got the following problem for my Calculus course: http://img291.imageshack.us/img291/7449/problemsw3.jpg I know the answer to this is 2, but I can't seem to make any headway in this problem. Can somebody give me a hint?

Here is my problem: I need to integrate: (\frac{sin \alpha z}{\alpha z})^2\frac{\pi}{sin\pi z} around a circle of large radii and prove: \sum_{m=1}^\infty(-1)^{n-1} (\frac{sin m\alpha}{m\alpha})^2 =\frac{1}{2} I'm kind stumped. I've been looking at books for a while...

How many terms of : \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2} do you have to add to get an error < .01 Alright, I used the Alternating Series Estimation Theorem since the terms are decreasing and the terms approach 0. So, by the theorem, .01 < = b_{n+1} so 1/(n+1)^2 = 1/100...

Hey, I'm having a bit of difficulty with a specific problem. I was able to somewhat easily work the problem out, but the few different answers I've tried have been incorrect. I'm beginning to think there's something wrong with my trigonometry or something because I swear the calculus is...

The sum of a series: \sum _{n=0} ^{\infty} \frac{2^{2n+1}x^{2n}}{n!} is: a)2cos(2x) b)cos(x^2) c)e^{2x} d)2e^{2x^2} e) None of the above.I have absolutely no idea how I would go about solving this. I know various tests for convergence and divergence, but the only methods I have to calculate sums...

I have the two series: C = 1 + (1/3)cosx + (1/9)cos2x + (1/27)cos3x ... (1/3^n)cosnx S = (1/3) sinx + (1/9)sin2x ... (1/3^n)sinnx I have to express, in terms of x, the sum to infinity of these two series. Here's what I've done so far: Let z represent cosx + jsinx C + jS = z^0 +...

i need a little help with this problem: determine if the infinite series converges or diverges. summation (from n=1..infinity) {1/(n^2+i^n)} I first applied the ration test to this series and got (n+1)^2 + i^(n+1) / [n^2 + i^n] i then multiplied top and bottom by (n^2 - i^n) which...

Hello, I could use a big helping hand in trying to understand an example from a text. Let's say I have a convergent series: S=\sum_{n=1}^{\infty}{a_n} Okay, so now: \sum_{n=1}^{\infty}{a'_n} is a rearrangement of the series where no term has been moved more than 2 places. So, the exercise...

Hi all! Here's something I'm having difficulty seeing: Suppose u_n > 0 and \frac{u_{n+1}}{u_n} \leq 1-\frac{2}{n} + \frac{1}{n^2} if n \geq 2 Show that \sum{u_n} is convergent. I'm not sure how to apply the ratio test to this. It looks like I would just take the limit. I get: lim_{n...

Hello all! I have the following infinite series: \frac{10}{x}+\frac{10}{x^2}+\frac{10}{x^3}+\ldots How would I find a function, f(x), of this series? I know the series converges for \vert x \vert > 1 I think the function is: f(x) = \frac{10}{x-1} but I'm not sure how to get it. Thanks, Bailey

Hi, I'm working on this problem: Prove that if |x|<1, then 1 + x^2 + x + x^4 + x^6 + x^3 + x^10 + x^5 + ...= 1/(1-x). I know that this is a rearrangement of the (absolutely convergent) geometric series, so it converges to the same limit. My trouble is proving that the rearrangement...

Hi, I'm working on this problem: If {s_n} is a nondecreasing sequence and s_n>=0, prove that there exists a series SUM a_k with a_k>=0 and s_n = a_1 + a_2 + a_3 + ...+ a_n. I'm not sure where to start. I wrote out the sequence's terms: s_n = (s_1, s_2, s_3, ...s_n) Then I wrote...

Hello, Here's the question: Does the series SUM log(1+1/n) converge or diverge? I wrote out the nth partial sums like this: log(1+1) + log(1+1/2) + log(1+1/3)+...+log(1+1/n) It looks to me like the limit of the thing inside the parentheses goes to 1 as n goes to infinity, making the...

I am having trouble evaluating the sum \sum_{i=1}^{\infty}\frac{i}{4^i} by hand. My TI-89 is giving me an answer of 4/9 or 0.44 repeating, but I am uncertain how to go about solving this by hand and proving the calculator's result. To my knowledge, no identity or easy quick fix like the...

Okay, I have this function defined as an infinite series: f(x) = \sum_{n=1}^{\infty}\frac{\sin(nx)}{x+n^4} which is converges uniformly and absolutely for x > 0. I have shown that f is continuous and has a derivative for x > 0. Now I have to show that f(x) \rightarrow 0 as x \rightarrow...

I'm having a little trouble with the following problem. here it is... Determinte whether a_n is convergent for... a_n = \frac{2n}{3n+1} How would i go about solving this? Can i just simply take the limit and use L'Hospital's rule to see if it diverges or converges? Or is it a little...

Dear people of Physics Forums, I would like to have your opinion about the following problem: Suppose we are interested to evaluate the average efficiency ( e ) of an infinite series of carnot cycles. The temperature of the hot reservoir of each carnot Cycle of the series, Ta, is at 990K, but...

Is it generally acceptable to write the following: a_0 + \sum_{n=1}^{\infty}a_{-n} + \sum_{n=1}^{\infty}a_{n} as \sum_{n=-\infty}^{\infty}a_{n} ?

hey, i was just wondering if anyone could help find the sum of the infinite series defined by 1/[n(n+1)(n+2)]. I can split it into partial fractions but not sure from there. Thanks

Hey there, I'm working on a perturbation theory problem, and I have no clue where to start in solving an infinite series. It's an infinite square well with a delta function potential in the centre and I'm trying to find the 2nd order energy correction to Energy En. Anyway, what I've got is...

1+(1/2)+(1/3)+(1/4)+(1/5)...+(1/n) can sum one prove this series is divergent? or just tell me what the expression for sum to infintiy in terms of n is?

Today we started on infinite series, I'm getting the material just fine and able to do most of the problems, but one is giving me problems. \lim_{n\to{a}} 2n/\sqrt{n^2+1} I recognized that \infty/\infty so I can use L'Hopital's rule. So taking the derivative of the numerator and...

I'm reading How to Ace the Rest of Calculus and on page 31 there's a test for divergence that says if the limit (as n goes to infinity) of a_n is NOT equal to zero, then the infinite series \sum a_n (that goes from n = 1 to infinity) diverges. 3 pages before that, on 28, there's an example on...

The following series can be shown to converge, but exactly what does it converge to? Euler was supposed to have proven it to sum to pi^2/6, but how? 1 + 1/4 + 1/9 + 1/16 + ... + 1/(r^2) as r -> infinity The following is a small maths puzzle that was asked in another forum, but which I...

\textrm{(a) A sequence} \left\{ a_n \right\} \textrm{is defined recursively by the equation} a_n = \frac{1}{2} \left( a_{n-1} + a_{n-2} \right) \textrm{for} n \geq 3 \textrm{, where} a_1 \textrm{and} a_2 \textrm{can be any real numbers. Experiment with various values of} a_1...

In the first part of the question, I proved that \int_{1/2}^{2} \frac{ln x}{1+x^2} dx = 0 Then I needed to evaluate the following but I didn't know how to do it. Can you give me some clues? I know it must be related to the definite integral that I proved in the first part, but how...

hi, i need a little help calculating the infinite series sorry if it seems confusing, but i don't know how to put in the sigma or intergral symbols i did my best to make it clear: i am sopposed to express (integral from 0 to 1) of 2dx/[(3x^4)+16] as a sum of an infinite series here's what i...

is sqrt() also an infinite series? any other defination of sqrt() other than x^0.5?

Hi all! I was wondering which method one should use to find the actual sum of an infinite series. I know how to find the sum of a geometric series (if it converges), but how could I find the sum for, for instance \sum_{n=0}^\infty\left(\frac{n+5}{5n+1}\right)^n I know that it converges...

Where could i find a good introductionto renormalization theory ? ( i have a degree in physics but i do not know about renormalization). In fact i have some questions: Let us suppose we have the series: f(g)=a0+a1g+a2g**2+.. where g is the coupling constant and a0,a1,a2,a3..an are numerical...