how do you prove 1+ 1/2 + 1/3 + .... + 1/1000 < 10 ?

how do you prove 1+ 1/2 + 1/3 + .... + 1/1000 < 10 ?

There are many methods. My personal favourite is to approximate it by an integral.

1 + 1/2 + 1/3 + 1/4 + 1/5 .... < 1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 ..... < 1 + [1/2 + 1/2] + [1/4 + .... < 10

Sum of a series? Look in your formula book

I doubt this series will be in the formula book?

Unless your booklet had Euler's constant?

Really? 1/3 is smaller than 1/2

Really? 1/3 is smaller than 1/2

Do you mind explaining what you're trying to imply?

?

???

1 + 1/2 + 1/3 + 1/4 + 1/5 .... < 1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 ..... < 1 + [1/2 + 1/2] + [1/4 + .... < 10

Your series is less than:

1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + 1/8 + 1/8 + ...

Which is 1 + 2(1/2) + 4(1/4) + 8(1/8) + ... + 512(1/512)

Which is 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Which is 10

So your series is less than 10


I haven't explained it in the best possible way, but you see what I mean...