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

BowToTheFlyingCircus

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

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Reply 1

BowToTheFlyingCircus

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

Count them on your fingers - (Have you heard of the harmonic series? I suggeest you look this up on google).

Reply 2

Sum of a series? Look in your formula book :biggrin:

:biggrin:

Reply 3

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

Reply 4

SimonM

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

:yep:

Probably easiest method

Reply 5

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

Reply 6

Oops, I misread the question the first time.

Try to rewrite as the sum of lots of sums to 1. If you can show that this sequence is less than 10 of these sums, you're done.

Spoiler

Reply 7

n1r4v

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

Really? 1/3 is smaller than 1/2

Reply 8

...

Reply 9

Smiling...Rain

Sum of a series? Look in your formula book :biggrin:

:biggrin:

I doubt this series will be in the formula book?

Unless your booklet had Euler's constant?

EDIT: I think Simon's suggestion is the easest way.

Reply 10

DeanK2

I doubt this series will be in the formula book?

Unless your booklet had Euler's constant?

Even if it does. There is no closed form for the sum of the harmonic series (as far as I'm aware)

Reply 11

SimonM

Really? 1/3 is smaller than 1/2

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

Reply 12

SimonM

Really? 1/3 is smaller than 1/2

?

:smile:

Reply 13

n1r4v

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

No idea, was on another planet

Reply 14

Agrippa

?

:smile:

???

Reply 15

DeanK2

???

???

:p:

Spoiler

Reply 16

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...

Reply 17

BowToTheFlyingCircus

OP

n1r4v

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

Don't understand why you can do that? and how do you then know if its less than 10?

Reply 18

The integral method (by comparing areas)

\displaystyle \int_1^{1001} \frac{1}{x} \, dx < \sum_{n=1}^{1000} \frac{1}{n} < 1+\int_1^{1000} \frac{1}{x} \, dx

Which shows that it is

\displaystyle6.90875... < \sum_{n=1}^{1000} \frac{1}{n} < 7.907755...

I expect I've made an OBO

Also

\displaystyle \sum_{n=1}^{1000} \frac{1}{n} \approx 7.4854708605503449126565182043339001765216791697088

Reply 19

tazarooni89

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...

I've seen this method before, and (I think) I follow it, but how do you know when/how to do that?

Like, where do you pull the 1/8 from? Obviously it's 2^3, but why is it significant?

If I got this question in a test/interview, I'd never think to do that.

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