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Question

Use given problem to get good upper and lower bounds for the sum of the first 10 million terms of the harmonic series.

Solution

VerifiedAnswered 1 year ago

Answered 1 year ago

Step 1

1 of 2On Problem 35 we have shown that for each natural number $n$,

$\ln\left(n+1\right)<\displaystyle\sum_{k=1}^n\displaystyle\frac{1}{k}<1+\ln\left(n\right).$

Thus, taking $n=10,000,000$ we have that

$16.1181\approx\ln\left(10,000,001\right)<\displaystyle\sum_{k=1}^{10,000,000}\displaystyle\frac{1}{k}<1+\ln\left(10,000,000\right)\approx 17.1181.$

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