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

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On Problem 35 we have shown that for each natural number nn,

ln(n+1)<k=1n1k<1+ln(n).\ln\left(n+1\right)<\displaystyle\sum_{k=1}^n\displaystyle\frac{1}{k}<1+\ln\left(n\right).

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

16.1181ln(10,000,001)<k=110,000,0001k<1+ln(10,000,000)17.1181.16.1181\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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