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Calculus IIChapter 4: Series and SequencesSection 4-5

Special Series — Question 1

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Question 1

Consider the Basel series ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}.

  1. Prove that the series converges using the pp-series test.

  2. State Euler’s classical evaluation of its exact sum and give a decimal approximation.

  3. Use an integral estimate to bound the remainder after NN terms, and explain why convergence alone does not produce the exact constant.

Original worksheet page 1: question and worked solution for 4-5-001
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Question 1 – Solution

Step 1: Classify the series.

This is a pp-series ∑1/np\sum1/n^p with p=2>1p=2>1. Therefore it converges. This test establishes existence of a finite sum but does not determine that sum.

Step 2: State the classical special value.

Euler’s solution of the Basel problem gives the non-elementary identity ∑n=1∞1n2=ζ(2)=π26≈1.644934.\sum_{n=1}^{\infty}\frac1{n^2}=\zeta(2)=\frac{\pi^2}{6}\approx1.644934. The equality with π2/6\pi^2/6 requires additional machinery, such as Euler’s product for sin⁡x\sin x or Fourier series; it does not follow from the pp-series test alone.

Step 3: Quantify convergence.

Because f(x)=1/x2f(x)=1/x^2 is positive and decreasing, ∫N+1∞dxx2≤RN≤∫N∞dxx2,\int_{N+1}^{\infty}\frac{dx}{x^2}\le R_N \le\int_N^{\infty}\frac{dx}{x^2}, so 1N+1≤RN≤1N.\frac1{N+1}\le R_N\le\frac1N. Thus the partial sums approach π2/6\pi^2/6 from below, with an error of order 1/N1/N.

Original worksheet page 2: question and worked solution for 4-5-001

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