1. Consider the series ∑_{i=1}^{∞} a_{i}(−1)^{i+1} where for all i we have a_{i} ≥ 0 and the sequence (a_{i}) converges to 0. Prove that the series is convergent. Hint: Let s_{m} = ∑_{i=0}^{m} a_{i}. Prove that (s_{2n} is monotone increasing and bounded above by a_{1}. Do something similar with (s_{2n−1}. Then apply a theorem to each of these two sequences, and then stitch the results together somehow.

2. Consider the sequence (t_{i}) where

t_{m} = ∑_{i=1}^{m} 1/i − ∫_{1}^{m} x^{−1}dx.

Show that t_{n} > 1/n > 0∀n ∈ ℕ, and moreover that t_{n} > t_{n+1}∀n ∈ ℕ. Conclude that the sequence whose n-th term is

1 + 1/2 + 1/3 + … + 1/n − log n

is convergent. (The limit is often called γ, or Euler’s number or Euler’s constant).