I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the third one would go to 1. Can someone please confirm that this is in fact correct?


Question: does re-encrypting the same value with multiple keys reduce security
Answer: The "security" of an encryption scheme is commonly defined through a so called indistinguishability games, i.e. the attacker picks two messages of same length. You pick one of those two at random, encrypt it, and give it to the attacker. 
If the scheme is "secure", then the attacker's advantage of guessing which messages was encrypted should not be negligibly bigger than pure guessing, i.e. 1/2. This kind of security notion comes in different flavours like CPA, CCA1, and CCA2.

If a private or public key encryption scheme is secure against CPA, CCA1, or CCA2, then it will also be secure against the same games, where instead of giving two messages, the attacker gives you two message vectors and you encrypted one of them. This is proven with a standard technique called hybrid argument, which basically shows that if you can win a game with vectors of messages, then you can also win it where the attacker only outputs two messages.
Very roughly speaking you create a reduction, where your attacker guesses a position in the vector and inputs the challenge from the two message game. The probability of guessing the "correct" position in the vector is non-negligible if your vector is of polynomial length. There are plenty of books (e.g. the one by Katz &amp; Lindell) or online resources that explain this in depth, so I won't reiterate it here.

In short: If something is secure w.r.t. to for instance CPA, then it will also be secure if you obtain encrypted message vectors (or as you state where one message is encrypted under multiple different keys). The success-probability might change, but only by a very small (still negligible) amount, so nobody cares, since it's still secure w.r.t to the notion.