Recent content by Esseintes

Homework Statement Suppose the real power series \sum ^{\infty}_{n=0}c_{n}x^{n} has radius of convergence R > 0. Define f:= \sum ^{\infty}_{n=0}c_{n}x^{n} on I:= (-R, R) and let b \in I. Show that there exists a power series \sum d_{n}(x-b)^{n} that converges to f(x) for |x-b| < r - |b|...

The forum kept deleting my formatting so I put the tree in this picture. http://sites.google.com/site/blackburnt/_/rsrc/1238089640948/Home/tree.JPG Does this help?

Got it, thank you. I see now that it is much the same as showing that 1 + 1/2 + 1/3 + ... diverges.

Homework Statement Let {a_n} be positive, decreasing. Show that if a_1 + a_2 + a_3 + ... converges then lim n * a_n = 0. Homework Equations None. The Attempt at a Solution Consider the harmonic series 1 + 1/2 + 1/3 + ... . Observe that [a_n] / [1 / n] = n * a_n . Since 1 + 1/2 + 1/3 +...