The sum 1/2 + 1/3 + 1/4 + 1/5 + ... results in infinity...!

We know 1 + 1 + 1 + 1 + ..... reaches infinity.

So, in a similar way, 0.9 + 0.9 + 0.9 + 0.9 + ...... reaches infinity.

What about 1/2 + 1/3 + 1/4 + 1/5 + ....?

Do you believe it reaches infinity?

If yes, you can ignore the rest of the post.

If you want to try it yourself, you can close this page.

But if you do not believe it reaches infinity, you are not alone!

Warning: This post is about proving the sum as simply as possible. Please don't expect more ;)

How do we want to prove it?

By comparison.

Say we have the value A, with us.

Suppose we get another value B, with two conditions:-

A > B.
B reaches infinity.

So, we can conclude A reaches infinity!! Job done.

Splitting of our sum for comparing:-

Take the first terms of our sum -

1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10. Yeah, stop at 1/10.

Now, how many terms it has? 9. Now take another sum.

1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 (yeah, 9 times).

You can be sure about the second sum - It is 0.9

What about the first sum? Yeah we don't need an exact value, but we can observe something - It is more than 0.9 !!

Now, take the next terms in our sum - to say, the next 90 terms.

1/11 + 1/12 + 1/13 + 1/14 + ........... + 1/100

So, yeah, we need another 90 terms to match the above count -

1/100 + 1/100 + 1/100 + ........... + 1/100

Yeah, the second sum reaches 0.9, again. The first one? Something more than 0.9

Just as you might have understood, just repeat the terms in that way

1/101 + 1/102 + ... + 1/1000 and, the corresponding sum to compare would be a 900 times summation of 1/1000.

Result time!

As you observed, our second sum is taking the following shape -

0.9 + 0.9 + 0.9 + ......

But, we can write the sum at our hand as -

(more than 0.9) + (more than 0.9) + .....

So, yeah, if every term in the above sum is more than 0.9, and it is added for infinite times, the sum, of course, reaches infinity!!

Yeah, we are done!

Thanks for reading the post, and don't forget to share if there is someone you know who can enjoy reading this!

P.S. if you want to share any views/thoughts on this post, please do so in the comments section!!

Comments

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