Question

Что можно сказать о сходимости числового рядаc положительными членами ∑an, если известно, что ряд ∑2an сходится?

Answer 1

Как написано, двойку за скобки  и первый ряд сходится к половине предела второго ряда.

Answer 2

Convergence of a Series with Positive Terms

When considering the convergence of a series with positive terms ∑an, if it is known that the series ∑2an converges, certain conclusions can be drawn.

The convergence of the series ∑2an implies that the series ∑an also converges. This is due to the comparison test for convergence of series, which states that if 0 ≤ an ≤ bn for all n and ∑bn converges, then ∑an also converges. In this case, bn = 2an, and since ∑2an converges, it follows that ∑an also converges.

Therefore, if the series ∑2an converges, then the series ∑an also converges. This relationship is based on the comparison test for convergence of series.

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