In the first one , so... ?
The second one:
In the third: use Ratio test - Wikipedia, the free encyclopedia
show why the con/diverge:
i think she diverges. because the terms get bigger (closer to 1 each) so you are just adding almost 1 + 1 +1 +1 ... it tails off to infinity... diverge... but i think i need to use a rigurous proof not just intuition.
i'd say she converges since the denom gets rapidly massive so the whole thing is getting rapidly tiny... i can't see what it converges too though? and how to show it?
as above, i think its the same.
but i don't know how to show these using the whole episolon thing. the epsilon thing really bugs me. do i need the epsilon thing? can someone help me to show these things rather than just use intuition... also am i right or wrong???
In the first one , so... ?
The second one:
In the third: use Ratio test - Wikipedia, the free encyclopedia
A necessary (but not sufficient) condition for a series to converge is that the individual terms have to tend to 0.
Therefore, a valid way to show that a series diverges is to show that the individual terms do NOT tend to 0.
For the first
Clearly, the terms do not tend to 0, so the series diverges.
Think about it like this. Suppose you have some series. Since there may be some negative values in it, the sum will never be any greater than the sum of the absolute values of the terms (since they are all positive). Therefore, by the comparison test, if the "larger series" (the series of absolute values) converges, then so must the "smaller series" (the original series).
So for your second series, by showing that converges, you show also converges.
thats what we call nonnull test. so am i right there (do you think i would need to prove the sequence of terms converges to 1 or can i just state it doesnt converge to 0 by intuition?)
as for the 2nd one... ok i am using the comparison test, and the property of absolute convergence... yes this is one of the properties we are told. i think i get that one now but how do you know the example you gave is always greater (or equal) to the series in question?
my main problem is knowing what needs to be shown and what can just be stated... :S
see my intuition was correct
but i dont always know how to show it
of course i know this and i am an idiot for not seeing this was neccessary. thanks for the pointer
but wait doesnt imply anyway so why do i have to bother with using absolute convergence properties in the first place? can't i just directly use the comparison with the |1/5^r|?
also we would need to show that 1/5^r converges, but how?
(can someone just confirm my suspicion here also - that it is sinr^2 and not just sin r, actually makes no difference here? is this just an attempt to deceive?)