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1. [itex]\lim\sum\limits_{k=1}^{n}\frac{2k-1}{2^k}[/itex]. I see that this is [itex]=\lim\sum\limits_{k=1}^{n}\frac{k}{2^{k-1}}-1[/itex] but I have no idea what to do with that. I tried writing it as [itex]\lim\frac{\sum\limits_{k=1}^{n}2^{n-k}k}{2^{n-1}}-1[/itex] and then writing the top as [itex]x_n[/itex] where [itex]x_1=1, x_k=2x_{k-1}+k[/itex], but I failed. I hardly have any experience with recurrence relations.. And it seems overly complex.

2. [itex]\lim\prod\limits_{j=1}^{n}(1+\frac{aj}{n^2}), a\in\mathbb{R}[/itex]. (It might be that a can only be positive, I don't remember..). I have no ideas about this at all. The answer is [itex]e^{a/2}[/itex], I think.

A related question, is [itex]\lim\limits_{n\rightarrow\infty}(1+\frac{x}{n})^n=e^x[/itex] ? I don't see how that works..

Thanks for your time