section_path: Properties of series/Non-negative terms
section_name: Non-negative terms
paragraph_text: Non-negative terms
  
  When a is a non-negative real number for every n, the sequence S of partial sums is non-decreasing. It follows that a series Σ a with non-negative terms converges if and only if the sequence S of partial sums is bounded.
  
  For example, the series
  
  ∑ n = 1 ∞ 1 n 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}
  
  is convergent, because the inequality
  
  1 n 2 ≤ 1 n − 1 − 1 n, n ≥ 2, {\displaystyle {\frac {1}{n^{2}}}\leq {\frac {1}{n-1}}-{\frac {1}{n}},\quad n\geq 2,}
  
  and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the Basel problem.