Determine whether the following series is convergent"

1. a1= 2, an+1 = (5n+1)/(4n+3)an
is Summation of an convergent?
and

2. a1=1, an+1 = 10sqrt(an)

is Summation (-1)^n (100-an)^2 convergent?

Question

Determine whether the following series is convergent" 

1. a1= 2, an+1 = (5n+1)/(4n+3)an 
is Summation of an convergent? 
and 

2. a1=1, an+1 = 10sqrt(an)

is Summation (-1)^n (100-an)^2 convergent?

anonymous · Answer

Is the an in the first problem an exponent?

anonymous · Answer

no its just next to the (4n+3) term

anonymous · Answer

Not sure Ive seen something quite like this. But if these obey the normal rules that Im used to series, I would think we could use some of the common tests like nth term, etc.

anonymous · Answer

Are you familiar with those tests, or is this something different?

anonymous · Answer

i am, but im having trouble seeing how to apply them. this is a rly difficult question

anonymous · Answer

Alrighty. Well, since the an is there as an actual term, we can say that we have a series with an n^2 on bottom vs only an n^1 on top. If I ignore constants, I essentially have n/n^2 = 1/n. So it's reasonable to use 1/n in a comparison test.

anonymous · Answer

ok

anonymous · Answer

so then it would diverge

anonymous · Answer

Right, because 1/n is a divergent series and the an is greater than 1/n, which forces an to diverge.

As for the second question, this is simply nth-term test. Whenever the an of the series does not have a limit as n goes to infinity of 0, we can immediately say it diverges.

anonymous · Answer

As for the last one, we have an alternating series. So the conditions for an alternating series to converge is that an must have a limit of 0 as n goes to infinity, which we already fail right off the bat. With only a numerator termand no denominator, as n goes to infinity so does the an, so diverges for this one aswell.