Convergent Definition and 330 Threads

[Solved] Radius of Convergent Homework Statement Find the radius of convergent for \sum_{n=1}^\infty (1-2^n)(ln(n))x^n Homework Equations \frac {1}{R} = L = \lim \frac{a_{n+1}}{a_n} The Attempt at a Solution lim \frac {(1-2^{n+1})(ln(n+1)}{(1-2^n)(ln(n))} = L lim...

Homework Statement Decide if \sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n+1}-\sqrt{n}}{n} is convergent and if it is, is it absolutely convergent or conditionally convergent? The Attempt at a Solution I'm pretty sure that the \lim_{n\rightarrow\infty} a_n = 0 Am I supposed to use...

[SOLVED] Counterintuitive Convergent Series Homework Statement One of my new textbooks in mathematical analysis makes a very strange claim (not sure if it was a true claim or some random historical anecdote) for a convergent series in one of its short sections on the history of mathematics...

Prove that \sin (n^2) + \sin (n^3) is not a convergent sequence.

Homework Statement Determine whether the series \sum_{n=2}^{\infty}a_n is absolutely,conditionally convergent or divergent a_n=\frac{(-1)^n}{\sqrt{n}(\frac{2n}{n+1})^\pi} The Attempt at a Solution from Abel's test.c_n=\frac{(-1)^n}{\sqrt{n}}is convergent.and...

Homework Statement Let Sumation "a sub n" be an absolutely convergent series, and "b sub n" a bounded sequence. Prove that sumation "a sub n"*"b sub n" is convergent. (sorry fist time on this site and can't use the notation.) Homework Equations Theorem A: that states sumation "a sub n"...

How can you deduce that nad bounded sequence in R has a convergent subsequence?

Hello, I have a question i can figure out. THE QUESTION: Show that \sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, ... converges and find the limit.From what I see, the first term is root 2, the second term will be the root of 2 times the first term making it larger than the first...

Homework Statement Let {x_n} be a sequence in a metric space such that the distance between x_i and x_{i+1} is epsilon for some fixed epsilon > 0 and for all i. Can it be shown that this sequence has no convergent subsequence? Homework Equations None. The Attempt at a Solution...

Homework Statement How do you show a sequence of functions in terms of n is convergent (in general)? The Attempt at a Solution Do you presuppose the value of the function say v then show d(fn,v)->0 for n large? v is determined by looking at the sequence of functions and guessing...

Following Euler if we define the product: (x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s})...=f(x) taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that f(x,s)=1/Li_{s} (x) (inverse of Polylogarithm) however I'm not 100 % sure, although for x=1 you get the inverse...

Consider the series with ascending (but not necessarily sequential) primes pn, 1/p1+1/p2+1/p3+ . . . +1/pN=1, as N approaches infinity. Determine the pn that most rapidly converge (minimize the terms in) this series. That set of primes I call the "Booda set."

The problem states: Suppose \sum a_n and \sum b_n are non-absolutely convergent. Show that it does not follow that the series \sum a_n b_n is convergent. I tried supposing that the series \sum a_n b_n does converge, to find some contradiction. So the series satisfies the cauchy criterion and...

Attahced is a file of a problem I am trying to solve. Thanks for any help

This is a question on a recent assignment that I can't figure out. I think if I understood the first part, I could get the rest. Let {a_n} be a convergent sequence with limit L. Prove or provide counter examples for each of the following situations. Suppose that there exists a number N...

I am doing research into brain systems. Does anyone have any examples or links to existing equations where a a convergent matrice is the reverse of a divergent constant, such as log or phi ?

I want to know if the integral \int_0^{\infty} dx/(4x^3 + x^(1/3)) is convergent or divergent?Thanks

Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound: Xn=(-1)^n could anyone help?! thanks in advance!

(This isn't homework :redface:) Does this series converge? {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} } Does this series converge? {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} }...

I just have a quick question, is cos and sin divergent or convergent? I keep getting mixed results from different sources. I know that both functions oscillate so on the interval [0, infinity) they both diverge. But for some of my homework problems relating to improper integrals, the book...

I found this in another threat however i do not know wat he means by convergent sequences. Is something like when u trying to take the limit at an ASYMPTOTE of a fuction? i know that the limit doesn't not exist( or goes to infinitive i cannot recall) is that wat he means by convergent sequence...

Given this sum s = \sum_{k = 1}^{{\frac{x}{j}} - 1} k^{n}j^{n+1} x and n are constants and x/j is a positive integrer and k is an integrer To what value s converges as {j}{\rightarrow}{0} ? Edit: I have found that the awnser is \frac{x^{n+1}}{n+1}, but i do not know how to obtain this...

Hi, Here is the question: Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity. To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner. My thoughts are: every convergent sequence has a convergent...

hello all well i think I am kind of brain dead, iv been workin on a lot of problems over the last few days, I can't see anything obvious anymore, well this shall be the last one for today (i hope), anyway here it is, suppose that for some x\not= 0 , the series \sum_{n=1}^{\infty} a_n...

hello all iv been workin on this problem its kind of awkward check it out {an} is a decreasing sequance, an>=0 and there is a convergent series Sn with terms an we need to prove that the limit of nan is 0 i first started of a sequence bn=an+1+an+2+...+a2n then I showed that the limit...

Does anyone have tips on how to sum the following series? \sum_{n=1}^{\infty} n^2 w^n Region of convergence is for |w| < 1

In the harmonic series 1+1/2+1/3+1/4+... we omit expressions which contain digit 9 in denominator (so we omit e.g. 1/9, 1/19, 1/94, 1/893, 1/6743090 etc.). Proof that such series is convergent. Have You got any idea how to solve this problem? Thanks a lot for help

\int_9^{inf} \frac{1}{x^{6/5}} first thing i did was found the integral of the function \frac{5}{x^{-1/5}} then plug in inf(i will name it b) and 9 \frac{5}{b^{-1/5}} - \frac{5}{9^{-1/5}} now i will find the lim -> inf well for \frac{5}{9^{-1/5}}, it's equal to 7.759 now for...

Consider the following statement: If \left\{ a_n \right\} and \left\{ b_n \right\} are divergent, then \left\{ a_n b_n \right\} is divergent. I need to decide whether it is true or false, and explain why. The real problem is that I checked the answer in my book; it's false, but I...

If you want to calculate the electric field at a distance r from a line of infinite length and uniform charge density you could one of three things: 1. Employ symmetry and Gauss' law. 2. Use superposition and integrate from minus to plus infinity along the rod. 3. Integrate to find the...