







Convergent or divergent? 
20090303 

From Betsy: Use the ratio test to determine if the series is convergent or divergent.
1+1/2^2+1/3^3+1/4^4
I know the general terms are a(n)=1/n^n and a(n+1)=1/(n+1)^(n+1)
but I can't simplify r=lim(n>inf.)=n^n/(n+1)^(n+1) Answered by Robert Dawson. 





Sigma from 0 to infinity of (n^3 / 3^n) 
20061115 

From Cedric: I'm wondering how you would find if this series converges or diverges?
Sigma from 0 to infinity of (n^3 / 3^n)
Does the n^3 dominate, or does the 3^n dominate? What about higher powers like n^10 / 10 ^ n ? Which one would dominate then? Answered by Penny Nom. 





A sequence that converges to e 
20030316 

From Dane: Something I noticed fooling around with a calculator about 30 years ago. Considering e = 2.718281828459045.... Using Window's Calculator you will find 1.1^{11} = 2.8531167... 1.01^{101} = 2.731861... 1.001^{1001} = 2.71964085... 1.0001^{10001} = 2.71841774... 1.00001^{100001} = 2.7182954... 1.000001^{1000001}1 = 2.178231875... 1.0000001^{10000001} = 2.178219643... There apears to be a pattern. My conjecture is: 1.'infinite number of zeros'1^{1'infinite number of zeros'1} = e. Answered by Penny Nom. 





Harmonic numbers 
20030312 

From Becky: What can you tell me about the limit of harmonic numbers as it reaches infinity? Answered by Penny Nom. 





Power series representations 
20001127 

From Grace: Is there a systematic way of finding a power series representation of a function? I understand that you have to manipulate the function so that it is of the form 1/(1x), but beyond that I am lost. Answered by Harley Weston. 





Radius of convergence 
19990421 

From Nowl Stave: Why is the radius of convergence of the first 6 terms of the power series expansion of x^(1/2) centered at 4 less than 6? Answered by Harley Weston. 





A Series 
19990420 

From Deepak Shrestha: Given the sequence an=e^(n*Ln(n)), does the series converge and why? Answered by Harley Weston. 





Neighbourhoods 
19960214 

From Robert: What are neighbourhoods? Answered by Harley Weston. 

