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I'm trying to understand which is the sum of the following telescoping series (I showed this is converging, I'm not reporting here):

$$\sum\limits_{j=n}^{\infty} [\mathbb{P}(E_j) - \mathbb{P}(E_{j+1})]$$

My take is that every term except the first and last one are canceling themselves so the sum should be, if I'm correct, given by $\mathbb{P}(E_n) - \lim\limits_{j\to\infty}\mathbb{P}(E_j)$

Is this OK?

Baffo rasta
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  • Look at the partial sums, then take the limit. I believe you will get an "yes" to your question. – rtybase Dec 13 '18 at 08:16

1 Answers1

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Yes $$\sum_{j=n}^m(\mathbb P(E_j)-\mathbb P(E_{j+1}))$$ $$=\mathbb P(E_n)-\mathbb P(E_{m+1})$$ Applying $\lim_{m\to\infty}$ on both sides, we get, $$\sum_{j=n}^\infty(\mathbb P(E_j)-\mathbb P(E_{j+1}))$$ $$=\mathbb P(E_n)-\lim_{m\to\infty}\mathbb P(E_m)$$

Hope it is helpful

Martund
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