Recent content by anthony.g2013

The first few terms of power expansion of (1+1/n)^n would be e^(n(1/n -1/2n^2 + 1/3n^3-...))= e^(1- 1/2n +O(n^2)). Now the e terms will cancel and we will be left with e^(-1/2n+O(n^2)). How to proceed form here?

Okay so is it true that lim (1+1/n)^n as n→∞ is the same as ∑ (1+1/n)^n as n goes from 1 to infinity? If so, then ∑ e -e +c/n leaves us with the harmonic series c∑1/n which clearly diverges. So c=0. Please let me know if I am correct. If not, then you might as well give me an answer as all my...

Of course, if m=x, then we will have the limit as 1.

Okay let's do it the way you suggest. When m=1, x=1, then cos(pi)^2n=1 as n→∞. When m=1, x=1/3, then cos(pi/3)^2n= (sqrt3/ 2)^infinity = 0. So the two possible outcomes are zero and 1. So since m is an integer, it all comes down to x. If x=integer, then the limit is 1 and if x=/ an integer, then...

Nope. Why don't you suggest one and I can may be look it up?

Can't really think of anything else. Any suggestions?

Homework Statement Is there a real number c such that the series: ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent? The Attempt at a Solution I used the ratio test by separating each term of the function as usual to find a radius of convergence, but that doesn't...

Could you please show me how you would do the problem? I have tried a variety of things like converting cosine squared into sine squared, or using e^ln form, but nothing seems to work. I am sorry, I am completely blank.

I see your point. The limit varies as we vary m and x, but I am still stuck. So do we just say that there is no limit because the cosine function keeps oscillating between -1 and 1? I still have no clue how to deal with the m!

Actually I should clarify the question and mu doubt: the actual question is Evaluate lim m→infinity [lim n→ infinity ((cos (m!*pi*x))^2n) ]. So if we went with -1<= cos (m!*pi*x)<= 1, take powers of 2n on all sides and take limit as n goes to infinity, we get the limit of the original...

Homework Statement What would be the limit of {cos(m!*pi*x}^{2n} as 'n' and 'm' go to infinity along with proof? Also x is real. I have no clue as to how to start the question. Please help! Thanks. The Attempt at a Solution I tried to write the original function as...