Assume converges
where and for
show that: converges
Excuse the doubt, but all these "summon the heroes" exercises look suspiciously similar to rather standard exercises in calculus I and II...
Are you sure these aren't "summon the guys that are going to do my homework for me" exercises? I've seen questions asked by you some few weeks ago and they did look like homework and not summonheroeds...
Tonio
I only want to make some fun in solving math problems(like playing game)
do not finish my homework.....
actually ,I am not a student now...
Thanks very much
My idea is:
As converges
so I think for massive(not all) ,
then will give the conclusion
But now I must to prove some satisfy the condition: , do not
make any essential contribution to the summation
Note that is decreasing. Therefore, the problem is equivalent to:
If converges, where decreases, then converges.
(Applying this to gives the initial statement)
Using the property from your previous "challenge", the proof is then straightforward.
(You can also directly use this property without rewriting the statement: hence and , qed)