Harmonic series/Computation/Termination/Exercise

Karl wants to compute approximately with his programmable pocket calculator the value of the series ${}\sum _{n=1}^{\infty }{\frac {1}{n}}$. The display of the calculator shows ${}8$ positions after the decimal point. The calculator performs one addition (meaning a member of the series is added to the previous sum) per second and shows the result. Karl has programmed correctly and considers the following strategy: "If the display does not change for a complete hour, then we are probably close to the true result and we accept this as a good approximation. Then the calculator shall stop“.

1. What is roughly the last member to be added?
2. What do you think about this strategy?

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