Alternating series Definition and 109 Threads

I was just thinking about the following series: 1-2+3-4+5... I'm not familiar with any other series like this one(other than the alternating harmonic), and I was curious as to whether or not it would be convergent, and if reordering it to -1+2-3+4-5... would change its convergence...

Homework Statement given an = ( -1 )(n+1) / \sqrt{n} determine if the infinite series is Absolutely Convergent, Conditionally Convergent, or Divergent. Homework Equations I hope I have these theorems down correctly, please correct me if I'm wrong. If \Sigma|an| is Convergent then...

I am looking at the following problem: sum from n=1 to inf: (-1)^n * n/(n+2) I used the alternating series test on this, with an = n/n+2 and a(n+1)=(n+1)/(n+3) lim an = 1, not equal to zero, so I had the series diverging. The book simply said lim n-> inf of |an| =1 Is my way right? If...

I have a specific problem but more than figuring out the answer I just want to figure out how to deal with factorials. My book is less than helpful on it... The problem is... \sum_{n=1}^{\infty} (-1)^n \frac{n^n}{n!} I understand that I have to take the limit of the sequence...

There's something that's confusing me about what appears to be the standard form of stating the alternating series test in basic calculus. The four sources I looked up were James Stewart's CALCULUS, Howard Anton's CALCULUS, wolfram alpha's mathworld, and wikipedia. All four had essentially the...

Homework Statement Sorry I don't know how to use symbols on this site so bear with me: the question is does the following series converge: sum of sin(1/n^2) where n goes from 1 to infinity Homework Equations Limit comparison test, maybe others The Attempt at a Solution Okay I think I got...

The definition I am working from is "Let Z=(z(sub n)) be a decreasing sequence of strictly positive numbers with lim(Z)=0. Then the alternating series, Sum(((-1)^n)*Z) is convergent. My question is how to solve the following: If the hypothesis that Z is decreasing is dropped, show the...

Homework Statement test for convergence by alternating series test 1. (∞, n=1) ∑ ((-1)n+1 n2)/(n3 + 4) 2. (∞, n=1) ∑ (-1)n sin(pi/n) Homework Equations alternating series test bn = |an| alternating series is convergent if satisfies 1. bn+1 ≤ bn for all n 2. lim n--> ∞ bn =0 The Attempt...

Homework Statement Determine the value of the sum \sum_{n=1}^\infty \frac{(-1)^n}{1+n^2} I have determined it trough the use of Fourier series, but does there exist another way to do it?

Homework Statement test the series for divergence or convergence, and the only thing we have learned so far is alternating series test (∞, n=1) ∑ (sin(nπ)/2)/n! Homework Equations The Attempt at a Solution So I know that the alternating series test has two conditions that need to be...

So I have this series: \sum^{infinity}_{n=3}(-1)^{n-1}\frac{ln(n)}{n} And I'm trying to use the AST to find out if it converges or not. First of all, I'm stuck trying to show that ln(n)/n is decreasing... But then after that. I'm assuming I can compare it with 1/n to show that it...

Is it possible for a series to converge without the constraint that a_n+1< or equal to a_n? Can we have a convergent series with only the requirement a_n >0 and the limit as x approaches infinity = 0 (i.e. not a decreasing monotonic series)? If yes include 3 series which disprove the...

Homework Statement The problem asks you if the series: 1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}... converges or diverges Homework Equations The Attempt at a Solution I tried to apply the Leibniz rule but I realized it can't be applied. Is there a transformation of this...

Homework Statement Given the following: 1 - e + e2/2! - e3/3! + e4/4! + ... Find the sum of series Homework Equations The MacLaurin equation for ex The Attempt at a Solution Well I thought that it would look like \sum(-1)^n\frac{e^n}{n!} Tried the Ration Test and got no where...

Homework Statement summation --> (-1)^n+1 (2/3)^n (I don't know how to do the symbol for sum) The Attempt at a SolutionI) lim n-->∞ (2/3)^n = limit does not exist ? It diverges ? P.S I am not sure if this is true. Any explanation will be a great help. Thanks.

The alternating series test states that if b[n] is decreasing and the lim n-->infinity of b[n]=0, then the series converges. However, does this also mean that if one of the two conditions is not satisfied, then the series is automatically divergent?

Homework Statement Ok, I know that (-1)^r \binom {n} {r} is supposed to equal 0. But I have plugged some numbers into this series, and this doesn't seem to be true for even numbers of n? Like for n = 4 and r = 4, I have: 1 - \frac{4!}{1!3!} + \frac{4!}{2!2!} - \frac{4!}{3!1!} +...

Homework Statement \sum_{n=1}^\infty(-1)^n \frac{n^n}{n!} Homework Equations I can not find my limit as n approaches infinity. I know that the answer is infinity but I am not sure how to get it. The Attempt at a Solution

Homework Statement Prove that 1 - \dbinom{n}{1}\ + \dbinom{n}{2} \ - \dbinom{n}{3} \ + \cdots \ + (-1)^r \dbinom{n}{r} = (-1)^r \dbinom{n - 1}{r} by induction. Homework Equations The Attempt at a Solution Well, I know how to solve the normal binomial sums by using the...

Consider the series (to infinity; n=0) (-1/3)^n (x-2)^n Find the intercal of convergence for this series. To what function does this series converage over this interval? I know this is an alternating series...I just don't know how to go about it. Thanks for you help. ** Should I...

Homework Statement My prof gave us an extra credit opportunity for a few extra points on the final exam(tomorrow). He told us to go find an example of an alternating series that is decreasing, its limit->0, and it diverges. So far I haven't seen any examples, plus I have sat around some...

The alternating series test contains two conditions for convergence. The first condition is that the nth term (extracting the power of -1) is always positive and monotonically decreasing. The second is that the limit of that nth term goes to 0 as n goes to infinity. I've seen a proof...

Homework Statement [PLAIN]http://img696.imageshack.us/img696/3438/46981606.jpg Homework Equations The Attempt at a Solution in those squars, I am sure about everything that i did and i get it wrong.. the only thing i don't know is bn+1 how would i know if its =, < or > than...

Homework Statement \Sigma(-1)^{n+1}\frac{1}{n!} How many terms will suffice to get an approximation within 0.0005 of the actual sum? Find that approximation. Homework Equations No idea.The Attempt at a Solution What I tried doing is setting my absolute value of the series less than 0.005, but...

i have a question regarding adsolute and conditional convergence of alternating series. - i know that summation of [ tan(pi/n) ] diverge, but how do we proof it converge conditionally? (ie, (-1)^n tan(pi/n) ] can Leibiniz's theorem be used in this case? but tan(pi/2) is infinite? any...

Homework Statement Okay, well this was a question on one of my recent tests: How many terms do you have to use to estimate the sum from n = 0 to n = infinity of (-e/pi)^n with an error of less than .001? Homework Equations Alternating series remainder theorem: For an...

I'm confused as to how to find an+1. I can figure out how to find the limit of an, which is the first component of the alternating series test. How do you find an+1? I'm thinking you just plug in "n+1" for "n" is that right?

Homework Statement What is \sum_{n=1}^{\infty} (-1)^n/n Homework Equations The Attempt at a Solution I know that the alternating series \sum_{n=1}^{\infty} (-1)^{n-1}/n converges to ln(2), but I am not sure how to find this series.

I need to test the series ∑_(n=1)^∞〖(-1)^(n+1) 1/(n+2x^2 )〗 x in R for absolute, pointwise and uniform convergence. I am unsure about absolute convergence. I thought about using the comparison test and the idea that 1/n+2x^2 > 1/n and so diverges. Is this correct? I'm not sure...

summation from 1 to infinite: (5n)/(n^2+1) What is the maximum amount of error of s4 as compared to the infinite series? error=S-Sn<bn+1 I got S4=-14/17 and S5=-25/26 but how you suppose to get S? i though you can't tell what a alternating series converge to

i am given te series: ∞ Σ(-1)n-1/{n+50*cos(n)} n=1 1st thing i check is lim |an| = 0 n->∞ so i know that the series might converge and might diverge, since this is a series with an alternating sign, i checked with lebniz rule, an>an+1 ==> not true for this series since...

Homework Statement Is this an alternating series : (-1)^n / ( arctan n ) ^ (n) Homework Equations I know that the alternating series should have (-1) to any power, but also the signs should be - , + , - , + ... or the opposit, The Attempt at a Solution I am ok with this I...

Homework Statement Calculate the sum of the following series: \sum_{i=2}^{\infty}(-1)^i \cdot \lg ^{(i)} n Where (i) as a super-script signifies number of times lg was operated i.e. \lg ^{(3)} n = (\lg (\lg (\lg n))) , and n is a natural number. Homework Equations The Attempt...

Homework Statement Determine the convergence or divergence of the series: series from n = 1 to infinity of cos (3npi). ((please rewrite in LaTex... idk how?)). The Attempt at a Solution I rewrote the series to the following: series from n = 1 to infinity of (-1)^n ... because the...

Homework Statement I have 2 McLaurin series. 1) ln(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... 2) ln (1-x) = -x - (x^2)/2 - (x^3)/3 - (x^4)/4 + ... The Attempt at a Solution I want to find the range of x values for which series 1) and 2) converge. For 1) I am using the...

Homework Statement Let s be the sum of the alternating series \sum(from n=1 to \infty)(-1)n+1an with n-th partial sum sn. Show that |s - sn| \leqan+1 Homework Equations I know about Cauchy sequences, the Ratio test, the Root test The Attempt at a Solution I really have no idea...

Can anyone help me out with calculating the partial sum of an alternating series? For example, how would I find the sum correct to 4 decimal places of: What I tried was finding how many terms it would take the have an error that was < .0001 then found the sum with that many terms... I...

Homework Statement Approximate the sum of the alternating series \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^4} by the 40th partial sum and estimate the error in this approximation. I know how to calculute the sum of the alternating series by 4th or 5th partial sum. I don't think this problem...

Homework Statement Find a sequence a_{n} which is non-negative and null but where \sum (-1)^{n+1} a_{n} is divergent. Homework Equations Alternating series test: Let a_{n} be a decreasing sequence of positive real numbers such that a_{n}\rightarrowa as n\rightarrow\infty. Then the...

Homework Statement Show that the series converges. Then compute an estimate of the limit that is guaranteed to be in error by no more than 0.005 \sum_{k=5}^{\infty} (-1)^k \frac{k^{10}}{10^k}The Attempt at a Solution This is obviously an alternating series and I know that C_{k} =...

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? The alternating series from n=1 to infinity of ((-1)^(n+1))/(n^2) |error| less than 0.0399 i got it down to 1/n^2 is less than 0.0399 but i can't figure out if n = 5 or n=6... please help:(

Find the sum of \Sigma^{\infty}_{k=1}\frac{(-1)^{k+1}}{k} I know the series converges because the coefficients go monotonically to zero. However it's been a few years since I was taught how to sum these series, so I'm having trouble. I thought about telescoping, but no terms seem to cancel each...

Homework Statement E(n = 1) to infinity ((-1)^n+1)/n^6 Homework Equations This needs to be in the proper form with the exponent on an being n - 1 not n + 1 The Attempt at a Solution I don't know how to get the problem into the proper for to evaluate it as an alternating series

Alternating Series and P-series "convergence" I couldn't resist trying out a pun. Anyway, onto the question: Homework Statement Test the series for convergence/divergence: \sum^{\infty}_{n=1}\frac{-1^{n-1}}{\sqrt{n}} Homework Equations Alternating Series Test and possibly p-series...

1. Homework Statement Determine the number of terms required to approximate the sum of the series with an error of less than .001 Sum ((-1)^(n+1))/(n^3) from n=1 to infinity 2. Homework Equations 3. The Attempt at a Solution I guess this is what you do: 1/(n+1)^3 <...

Homework Statement Determine the number of terms required to approximate the sum of the series with an error of less than .001 Sum ((-1)^(n+1))/(n^3) from n=1 to infinity Homework Equations The Attempt at a Solution I guess this is what you do 1/(n+1)^3 < 1/1000 and...

Homework Statement Determine wheter the series is convergent or divergent. If it convergent, approximate the sum of the series correct to four decimal places. heres the equation: http://img251.imageshack.us/img251/2261/46755781zg9.png Homework Equations The Attempt at a...

Homework Statement Test the series for convergence or divergence \sum\limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1} \frac{{\ln n}}{n}} Homework Equations If b_{n + 1} \le b_n \,\& \,\mathop {\lim }\limits_{n \to \infty } b_n = 0 then the series is convergent...

Homework Statement Test the series for convergence or divergence using the Alternating Series test: 1. sum of (-1)^n * n/(ln n) 2. sum of [sin(n*pi/2)]/n! Homework Equations The Attempt at a Solution 1. lim of n/(ln n) goes to infinity (as n->infinity), so it can't satisfy the...

I am having trouble finding \sum \frac{(-1)^n}{n^2} . (from n=1 to n=infinity) I know that \sum \frac{1}{n^2} is \frac{\pi^2}{6} . But the sum I need is slightly different than that, it is alternating... Any ideas/help would be appreciated! Thx