- #1

can something one tell me wheather it has a finit sum or not

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- Thread starter mathshead
- Start date

- #1

can something one tell me wheather it has a finit sum or not

- #2

Integral

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That is the http://www.jimloy.com/algebra/hseries.htm [Broken] series and it does NOT have a finite sum.

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- #3

STAii

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You know, when i was first told about the sum of an infinite geometrical series, it first looked impossible, then i was told "since the numbers are getting smaller and smaller, they add up to give a number (not infinity)".

And here the numbers are getting smaller and smaller, but still, they don't sum up to a number, why is this ?

- #4

mathman

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1/3+1/4>1/2

1/5+1/6+1/7+1/8>1/2

1/9+1/10+...+1/16>1/2

1/17+...+1/32>1/2

Keep this up and you get the harmonic series > 1+1/2+1/2+...

- #5

STAii

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My question may seem a little weird, but anyone that feels (s)he can help by even giving a hint would be great.

Thanks.

- #6

ahrkron

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Originally posted by STAii

...then i was told "since the numbers are getting smaller and smaller, they add up to give a number (not infinity)".

This is probably where your problem resides. The fact that the numbes get "smaller and smaller" is not enough to insure convergence, as you just witnessed. They need to get smaller "fast enough", so to speak.

- #7

selfAdjoint

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1 + 1/2 = 3/2

1 + 1/2 + 1/4 = 7/4

1+ 1/2 + 1/4 + 1/8 = 15/8

The partial sums are always of the form 2*2^n-1/2^n which is always less than 2, so the partial sums are bounded above and increasing, so they converge.

The harmonic series as the repeated proofs already posted show, doesn't do this, and this, not just the terms getting smaller is the true criterion for series convergence.

The terms getting smaller is a

- #8

STAii

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How exactly does it mean 'fast enough' ?

Is there somekind of relation that must be between each number and the number after it so that it has sum (i am not only talking about geometrical series).

Thanks.

Is there somekind of relation that must be between each number and the number after it so that it has sum (i am not only talking about geometrical series).

Thanks.

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- #9

Lonewolf

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- #10

Hurkyl

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The sum Σa(i) converges if and only iff:

lim

This is equivalent to Lonewolf's definition for real numbers. (the Cauchy criterion fails in incomplete metric spaces)

- #11

HallsofIvy

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The series [SIGMA] a

lim |a

- #12

ahrkron

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Originally posted by HallsofIvy

One idea of "how fast" numbers in an infinite series must get smaller is the "ratio test":

The series [SIGMA] a_{n}converges if

lim |a_{n}|/|a_{n+1}is less than 1

Just to clarify, I thik this is a sufficient condition, not a necessary one (1/n and 1/n^2 both fail the criterion, yet the latter is convergent).

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