Theorem. The Integral Test Estimate. Suppose that is a series which satisfies the hypotheses of the Integral Test using the function f and which converges to L. Let sn = a1 + a2 + ... + an be the nth partial sum and let rn = L - sn. Then

Discussion [Using Flash] [Using Java]

• Problem. Find an approximation of the series
  

  using the partial sum s₂₀. What is the maximum possible error using this approximation?
  Solution [Using Flash] [Using Java]

• Problem. Find an approximation of the series
  

  using the partial sum s₃₀. What is the maximum possible error using this approximation?
  Solution [Using Flash] [Using Java]

• Problem. Find an integer n such that, using s_n as an approximation of the series,
  

  the maximum possible error is at most .00001.
  Solution [Using Flash] [Using Java]

If we add s_n to each term of the error estimate given in the theorem above, we obtain the following which provides a way to obtain a better estimate for value of the power series.

Corollary.

• Problem. Using the Corollary, find an approximation for the series
  

  with n = 100 and find the maximum possible error in using this approximation.
  Solution [Using Flash] [Using Java]