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JH

# Use the following steps to show that the sequence $t_n = 1 + \frac {1}{2} + \frac {1}{3} + \cdot \cdot \cdot + \frac {1}{n} - \ln n$has a limit. (The value of the limit is denoted by $\gamma$ and is called Euler's constant.)(a) Draw a picture like Figure 6 with $f(x) = 1/x$ and interpret $t_n$ as an area [or use (5)] to show that $t_n > 0$ for all $n.$(b) Interpret$t_n - t_{n + 1} = \left[ \ln \left(n + 1 \right) - \ln n \right] - \frac {1}{n + 1}$as a difference of areas to show that $\left\{ t_n \right\}$ is convergent.

## a) $\left[\sum_{k=1}^{n} \frac{1}{k}\right]-\ln n>0$$t_{n}>0$b) Both the areas as $n \rightarrow \infty$ are then positive, and since we found that the difference in areas approaches a positive value for large $n, t_{n}-t_{n+1}$ must be $> 0$ and therefore $\left\{t_{n}\right\}$ is a decreasing sequence.c) convergent

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##### Kristen K.

University of Michigan - Ann Arbor

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Sequences

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##### Kristen K.

University of Michigan - Ann Arbor

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