Sir, an equation has no meaning for me unless it expresses a thought of GOD. - Srinivasa Ramanujan

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling". ...Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressons that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly...

If <math>n</math> is any positive quantity shew that
<math>\frac 1{n} > \frac 1{n+1} + \frac 1{{(n+2)}^2} + \frac 3{{(n+3)}^3} + \frac {4^2}{{(n+4)}^4} + \frac {5^3}{{(n+5)}^5} + \dots</math> Find the difference
approximately when <math>n</math> is great.
Hence shew that
<math>\frac 1{1001} + \frac 1{1002^2} + \frac 3{1003^3} + \frac {4^2}{1004^4} + \frac {5^3}{1005^5} + \dots < \frac 1{1000}</math> by <math>10^{-440}</math> nearly.