PlanetMath: limit of sequence as sum of series

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limit of sequence as sum of series

(Theorem)

If is the limit of a sequence

$\displaystyle u_1,\,u_2,\,u_3,\,\ldots$

can be expressed as the series sum

$\displaystyle U = u_1+\sum_{i=1}^\infty(u_{i+1}-u_i).$

Proof. Let $\displaystyle s_n := u_1+\sum_{i=1}^{n-1}(u_{i+1}-u_i)$ . We see that

$\displaystyle s_n = u_1+\sum_{i=1}^{n-1}u_{i+1}-\sum_{i=1}^{n-1}u_i = u_1+\sum_{j=2}^nu_j-\sum_{i=1}^{n-1}u_i = u_n$

for all $n = 1,\,2,\,3,\,\ldots$ Thus

$\displaystyle u_1+\sum_{i=1}^\infty(u_{i+1}-u_i) = \lim_{n\to\infty}s_n = \lim_{n\to\infty}u_n = U,$

Q.E.D.

"limit of sequence as sum of series" is owned by pahio.

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See Also: sum of series

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Cross-references: proof, sum, series, complex numbers, real, sequence, limit
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This is version 1 of limit of sequence as sum of series, born on 2007-08-12.
Object id is 9858, canonical name is LimitOfSequenceAsSumOfSeries.
Accessed 796 times total.

Classification:

40-00 (Sequences, series, summability :: General reference works )

Pending Errata and Addenda

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