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MATH
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Definitions from Infinite series

All definitions from Infinite series.

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Partial sums

For a sequence {an}\{a_n\}, the kk-th partial sum is

sk=∑n=1kan.s_k=\sum_{n=1}^{k}a_n.
  1. [1]:Infinite series

Infinite series

The series ∑n=1∞an\sum_{n=1}^{\infty}a_n converges to LL precisely when its partial sums satisfy

lim⁡k→∞sk=L,sk=∑n=1kan.\lim_{k\to\infty}s_k=L, \qquad s_k=\sum_{n=1}^{k}a_n.

In that case,

∑n=1∞an=L.\sum_{n=1}^{\infty}a_n=L.

The terms and the series answer different questions:

  • ana_n asks what happens to one term far out in the sequence;
  • ∑an\sum a_n asks what happens to the accumulated totals.

A convergent series must have an→0a_n\to0, but this is not sufficient: the harmonic series below is the standard counterexample.

  1. [1]:Partial sums
  2. [2]:Integral test
  3. [3]:Direct comparison test
  4. [4]:Limit comparison test
  5. [5]:Ratio test
  6. 8 more

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Partial sumsInfinite series