Question

How many terms `n` are required in order that the sum `sum_(k=1)^n 1/k` exceeds `100` ?

Answer 1

Approximately `1.5 xx 10^43`

Explanation:

The Euler Mascheroni constant is:

`gamma = lim_(n->oo) ((sum_(k=1)^n 1/k)-ln n) ~~ 0.5772`

For large values of `n` we have:

`sum_(k=1)^n 1/k ~~ ln n + gamma`

So an approximation to the required value of `n` would be:

`n ~~ e^(100-gamma) ~~ 1.5 xx 10^43`

or to the nearest integer above (in order to exceed 100):

`15092688622113788323693563264538101449859498`