If the positive-value sequences $\{x_n\}, \{y_n\}$ have limits 5 and 3 respectively, what are the limits of the sequences $\{x_n+y_{n+1}\}$, $\{x_ny_{2n}\}$, $\{2x_n\}$ and $\{x_n^{y_n}\} $.Enter your answer in the form *,*,*,*

Question

BlackTom AI · Accepted Answer

Consider the given limits: x_n -> 5 and y_n -> 3 with all terms positive.

First expression: x_n + y_{n+1}. Since y_n -> 3, shifting the index by 1 gives y_{n+1} -> 3 as well. Therefore, the limit of x_n + y_{n+1} is 5 + 3 = 8.

Second expression: x_n y_{2n}. Here y_{2n} also tends to 3 because the sequence y_n converges to 3, and evaluating at 2n does not change the limit. Thus the limit is 5 * 3 = 15.

Third expression: 2x_n. This is a constant multiple of x_n, so its limit is 2 * 5 = 10.

Fourth expression: x_n^{y_n}. With x_n -> 5 and y_n -> 3, and given that all terms are positive, the limit follows from continuity of the exponential function on positive bases: 5^3 = 125.

Putting all limits together in the requested order, the result is 8,15,10,125.

If the positive-value sequences \(\{x_n\}, \{y_n\}\) have limits 5 and 3 respectively, what are the limits of the sequences \(\{x_n+y_{n+1}\}\), \(\{x_ny_{2n}\}\), \(\{2x_n\}\) and \(\{x_n^{y_n}\} \).Enter your answer in the form ,,,简答题

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If the positive-value sequences \(\{x_n\}, \{y_n\}\) have limits 5 and 3 respectively, what are the limits of the sequences \(\{x_n+y_{n+1}\}\), \(\{x_ny_{2n}\}\), \(\{2x_n\}\) and \(\{x_n^{y_n}\} \).Enter your answer in the form *,*,*,*简答题