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\textbf{Math 166: Honors Calculus II}
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\textbf{Quiz 8}\quad {\sl March 30, 2000}

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\item Determine whether each of the following sequences converges or
diverges and find the limit of each convergent sequence.

  \begin{enumerate}[a)]

  \item $a_n = (-1)^n\Big(1 - \frac{1}{n}\Big)$

  \answerspace

  \item $a_n = \frac{\sin(n\pi/2)}{n\pi/2}$

  \answerspace

  \item $a_n = \Big(1 + \frac{c}{n}\Big)^n$

  \answerspace

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\item Let $0 < b < a$.

  \begin{enumerate}[a)]

  \item Prove that
%  $\lim_{n\to\infty} n^b\sum_{k=1}^n\frac{1}{(n+k)^{a+1}} = 0$.
  $\lim_{n\to\infty} n^b\sum_{k=1}^n\frac{1}{(n+k)^{\rlap{$\scriptstyle a+1$}}}
  \phantom{\scriptstyle a+1} = 0$.
  \qquad [ Hint: $\frac{1}{n+k} < \frac{1}{n}$ ]

  \answerspace
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  \item Prove that
  $\lim_{n\to\infty} n^a\sum_{k=1}^n\frac{1}{(n+k)^{\rlap{$\scriptstyle a+1$}}}
  \phantom{\scriptstyle a+1} = \frac{1}{a}\Big(1-\frac{1}{2^a}\Big)$.
  \qquad [ Hint: $\int_0^1 f(x)\,dx$~]

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