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# Logarithms Series Question watch

1. I have been working on this question:

Express the finite series

as a quotient of logarithms to base 2. Does the series converge as ?

I changed the base of each of the logarithms in the denominators to 2. This left me with the expression

.

I think that as the quotient will grow without bound as log is an increasing function. However I am not sure whether this is correct...
2. (Original post by FXLander)
I have been working on this question:

Express the finite series

as a quotient of logarithms to base 2. Does the series converge as ?

I changed the base of each of the logarithms in the denominators to 2. This left me with the expression

.

I think that as the quotient will grow without bound as log is an increasing function. However I am not sure whether this is correct...
Could also be written as

And yes, numerator is unbounded, denominator is a constant, hence off to infinity we go.
3. (Original post by FXLander)
I have been working on this question:

Express the finite series

as a quotient of logarithms to base 2. Does the series converge as ?

I changed the base of each of the logarithms in the denominators to 2. This left me with the expression

.

I think that as the quotient will grow without bound as log is an increasing function. However I am not sure whether this is correct...
Note that you can see that this diverges directly: suppose that and that we want to find . It is clear(*) that so that so once , we are adding a series all of whose terms are bigger than 1. So it diverges.

(*) If it's not clear, use specific numbers e.g:

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