Question

Need Help starting a calc question...

Answer 1

A series \[\sum_{?}^{?} a(n)\] is defined by the equations \[a(1) = 1\] and \[a(n+1) =a(n) (2+\cos n)/\sqrt{n} \], can someone help me start this question?

Answer 2

.... so write out the first few terms a_0, a_1, ...

Answer 3

first 3 terms I'm getting are 1+ (2+cos(2))+ (2+cos(2))(2+cos(3))/sqrt(3)

Answer 4

erm, sorry i got confused, the terms are 1+(2cos(1))+(2+cos(1))(2+cos(2))/sqrt(2) I believe

Answer 5

are you sure there's a nice solution? the first 10 terms are {1, 2 + Cos[
1], ((2 + Cos[1]) (2 + Cos[2]))/Sqrt[2], ((2 + Cos[1]) (2 +
Cos[2]) (2 + Cos[3]))/Sqrt[6], ((2 + Cos[1]) (2 + Cos[2]) (2 +
Cos[3]) (2 + Cos[4]))/(
2 Sqrt[6]), ((2 + Cos[1]) (2 + Cos[2]) (2 + Cos[3]) (2 + Cos[4]) (2 +
Cos[5]))/(
2 Sqrt[30]), ((2 + Cos[1]) (2 + Cos[2]) (2 + Cos[3]) (2 +
Cos[4]) (2 + Cos[5]) (2 + Cos[6]))/(12 Sqrt[5]), (1/(
12 Sqrt[35]))(2 + Cos[1]) (2 + Cos[2]) (2 + Cos[3]) (2 + Cos[4]) (2 +
Cos[5]) (2 + Cos[6]) (2 + Cos[7]), (1/(
24 Sqrt[70]))(2 + Cos[1]) (2 + Cos[2]) (2 + Cos[3]) (2 + Cos[4]) (2 +
Cos[5]) (2 + Cos[6]) (2 + Cos[7]) (2 + Cos[8]), (1/(
72 Sqrt[70]))(2 + Cos[1]) (2 + Cos[2]) (2 + Cos[3]) (2 + Cos[4]) (2 +
Cos[5]) (2 + Cos[6]) (2 + Cos[7]) (2 + Cos[8]) (2 + Cos[9])} ... doesn't look nice

Answer 6

Is there at least a nice way to tell if it converges or diverges? (sorry, I missed my lecture due to a snowstorm so i have no clue where to start with recursively defined sequences)

Answer 7

I would do that, but this series is defined recursively so I dont think that would work... Also the section that the question is in inplies you have to use the ratio test or the root test. Or determine i it converges absolutely.f

Answer 8

Yeah, same, but do i have to make it into a non-recursive series first?

Answer 9

euh mathematica tried, it's not pretty. {{a[n] -> (2^(1 - n) (E^-I)^(2 n) (E^(2 I))^n E^((I n)/2 - (I n^2)/2)
QPochhammer[E^I/(-2 + Sqrt[3]), E^
I, -1 + n] QPochhammer[-(E^I/(2 + Sqrt[3])), E^
I, -1 + n])/(Sqrt[Pochhammer[1, -1 + n]])}} what on earth is that, i have no clue

Answer 10

i think lim as n -> infinity of (a[n+1]/a[n]) = lim n->infinity (2+cos n)/sqrt(n) which sadly eventually goes to 0. so convergent.

Answer 11

But this is a series so it shouldn't be enough to say the sequence goes to 0 right?

Answer 12

but then what happens to the a(n) that is multiplied to the (2+cos n) / sqrt(n)? Do we just get rid of it?

Answer 13

the ratio test asks for a_n+1 / a_n.. so i just took it from it and took limits of both sides, i'm doing what it says ._.

Answer 14

If you are allowed to do that with a recursive sequence, thanks