Administrative: The VTRMC is soonish: Saturday 10/21, 9am to 11:30am, room W301. The Putnam is further off (Saturday December 2), but I need to submit a "team" (list of names) by 10/15. Today our focus is real analysis: limits, series, convergence. Basic things you should know: L'Hopital's rule. Standard power series (e^x, ln(x), sin(x), etc). Absolute convergence, radius of convergence. Estimate a sum by an integral. (See also: Euler-Mascheroni constant.) The sum of (1/n^s) diverges for s<=1, converges for s>1. Below I have collected example problems, mostly from the VTRMC, including two from the most recent year. Note that solutions to all the VTRMC exams are collected here: http://www.mathcs.emory.edu/~mic/math/files/vtrmc/solutions.pdf [VTRMC2016#1] Evaluate: Integral_{x=1}^2 (ln x)/(2 - 2x + x^2) dx [VTRMC2016#2] Determine the real numbers k such that Sum_[n >= 1] ((2n)!/(4^n n!n!))^k is convergent. [VTRMC2012#5] Determine whether the following series is convergent. Sum_[n >= 2] [ (1 / ln n) - (1/ ln n)^((n+1)/n) ] [VTRMC2011#5] Find the limit, as x approaches infinity, of (2x)^(1 + 1/2x) - (x)^(1 + 1/x) - x [VTRMC2010#7] Let Sum_[n>=0] a_n be a convergent series of positive terms (so each a_i > 0) and set b_n = 1 / (n a_n^2). Prove that Sum_[n>=0] [n / (b_1 + b_2 + ... + b_n)] is convergent. [VTRMC2010#6] Define a sequence by a_1 = 1, a_2 = 1/2, and a_(n+2) = a_(n+1) - (a_n a_(n+1))/2 Find the limit of n a_n, as n approaches infinity. [Putnam2016B1] Let x_0, x_1, x_2, ... be the sequence such that x_0 = 1 and for n >= 0, x_(n+1) = ln (e^(x_n) - x_n) Show that the infinite series x_0 + x_1 + x_2 + ... converges, and find its sum. Hints: [VTRMC2016#1] Substitute y = 2/x, you get a second integral that resembles I, take their sum. [VTRMC2016#2] Estimate each term as a power of n. Or, Stirling's formula. [VTRMC2012#5] Compare to Sum_n 1/(n ln n), which is known to diverge. [VTRMC2011#5] Show x^(1+1/x) - x is approximately ln x. I suggest power series (solution uses L'Hopital instead). [VTRMC2010#7] Rearrange (a_n) to be non-increasing. Then a_n <= SUM/n. [VTRMC2010#6] First show a_n decreasing to zero. Let b_n = 1/a_n, and show b_(n+2) - b_(n+1) has a constant limit. [Putnam2016B1] First show { x_i } is positive but decreasing. [ Full solution in 2016s.pdf ] Some additional problem files in today's directory (but without solutions): Series.pdf -- practice problems on series, not too bad limits.pdf -- a collection of Richard Stanely (hard to harder)