Summary of MENTOR problem-solving session
March 11, 2016

To view images full size, right-click on them and select "View image".

Problem 11.876

• Computing \(\sum_{k=0}^{n} \cos(kx)\) using the Euler formula for cosine and the sum of a geometric series; see answer in top left corner [parts crossed out in red were not a good approach]
  
  continuation, using the trigonemetric form of \(1 - (cos x + i \sin x)\) from below
  
• \(1 - (cos x + i \sin x)\) in trigonemetric form
  
• computations leading to it
  
• computing sums related to a geometric series; for the geometric series, see the above image (top right corner)
  since there is no closed-form expression for \(\sum_{k=1}^{n} \frac{1}{k}\), cannot expect to find a closed form for \(\sum_{k=1}^{n} \frac{x^k}{k}\) either
  but one can derive a closed-form expression for \(\sum_{k=1}^{n} k {x^k}\) and alike (use another calculus tool)
  

Problem 11.884

• try first a simpler problem: the first derivative instead of the second?
  
• So we would like an upper bound for \(\displaystyle \left(\int_0^1 \left(\int_{1/2}^t f'(x) dx \right) dt\right)^2\)
• the Cauchy-Schwartz inequality for sums
  there is also a version for integrals, might be useful here (at least, gives something related); note that this holds for any integration, need not be on \(\mathbb R^n\)
  actually, can take the absolute value inside the sum squared in the LHS (follows easily from the usual formulas)