Recent content by saddlepoint

Homework Statement A uniform rod of length l has an initial (at time t = 0) temperature distribution given by u(x, 0) = sin(\frac{πx}{l}), 0 \leq x \leq l. The temperature u(x, t) satisfies the classical one-dimensional diffusion equation, ut = kuxx The ends of the rod are...

Homework Statement Show if an > 0 for each n\inN and if ∑an converges, then ∑an2 converges and that ∑1/an diverges. NB. all ∑ are between n=1 and ∞ Homework Equations The Attempt at a Solution Let partial sums of ∑an2 be Sk = a1 + ... + ak To say ∑an2 is absolutely...

Ah right ok so by proving what's inside the brackets for {xn} ≤ 4 will be enough to prove that {xn} is bounded above. Then use of the monotonic sequence theorem shows {xn} is converging. Thanks for your help!

Thanks for your reply! I understand that the Monotone Convergence Theorem requires you to show {xn} is increasing and bounded above. However, I'm not sure how you can show it's bounded above without having to show the limit is exp(3).

Thanks for your reply! I'm not really sure how to try to show it's increasing the other way around. I find it difficult because it's obvious looking at it that n2 - n + 5 is increasing but proving this in a formal proof is hard. I don't think n > 0 necessarily so not sure where to start...

Homework Statement Show that {n2 - n + 5} is increasing and hence show that {xn} is convergent when {xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)] You may assume exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3. Homework Equations The Attempt...