Finding limit of S (1 Viewer)

[dan964]

It's a common trick to convert infinite products into infinite sums which are in some ways nicer to work with.

There were also two facts used that I didn't bother writing proofs of (you should try to make sure you understand why they are true):

1. log(1+x) =< x for all x > -1 where these things are defined.

This is an MX2 level application of calculus.

2. The harmonic series (1+1/2+1/3+...) diverges.

This can be done in many many different ways. If you like calculus, then you can bound the partial sums of this series below by an integral involving 1/x. The resulting quantity will grow like log(x) and hence diverge to +infinity.

when using the squeeze theorem what was the other function/limit you used??

 

[seanieg89]

when using the squeeze theorem what was the other function/limit you used??

P_N is always positive by definition, so we just need to bound it above by something that tends to zero.

In the notation I introduced earlier, this upper bound is:

C/exp(h_k-h_N)

where h_n = 1 + 1/2 + ... + 1/n denotes the n-th partial sum of the harmonic series, and C and k are constants.

(k and K mean the same thing, I changed my mind about capitalisation and couldn't be bothered to fix all of them.)