# Convergence theorem

1. Aug 6, 2013

### Miike012

Can someone explain to me what they are saying in the paint document? Because to me it seems like the statements are contradicting.

The first paragraph starts off with..." Let the fixed term be denoted...."

My concern is when the paragraph states.. "If the ratio is equal to unity, each of the succeeding terms is equal to u1 and the sum of n terms is equal to nu1hence the series is divergent.

First off that doesn't make sense and let me explain: Let the fixed term be denoted un and Let the infinite series be equal to
u1 + u2 + u3 + .... + un + .....

Now I will state what they said... If the ratio is equal to unity (un/un-1 = 1 or un = un-1) then each of the succeeding terms is equal to un and the sum of the n terms is equal to nun.

or in other words the series u1 + u2 + u3 + .... + un + .....
is equal to

nun + un+1 + un+2 + ......

However the first n terms which is equal to nun is a FINITE quantity and therefore the series will converge if the proceeding terms un+1 + un+2 + ...... converge or diverge if the proceeding terms un+1 + un+2 + ...... diverge.

Is this correct???

Another question:
Why do they say in the first highlighted portion that it will diverge but in the second highlighted portion say that it may or may not diverge?

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2. Aug 6, 2013

### haruspex

No, it's equal to u1 + u2 + u3 + .... + un + un + un + un + un + un + un + .....