If the positive-value sequences [math: {xn},{yn}]\{x_n\}, \{y_n\} have limits 5 and 3 respectively, what are the limits of the sequences [math: {xn+yn+1}]\{x_n+y_{n+1}\}, [math: {xny2n}]\{x_ny_{2n}\}, [math: {2xn}]\{2x_n\} and [math: {xnyn}]\{x_n^{y_n}\} .Enter your answer in the form *,*,*,*简答题
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Question

If the positive-value sequences [math: {xn},{yn}]\{x_n\}, \{y_n\} have limits 5 and 3 respectively, what are the limits of the sequences [math: {xn+yn+1}]\{x_n+y_{n+1}\}, [math: {xny2n}]\{x_ny_{2n}\}, [math: {2xn}]\{2x_n\} and [math: {xnyn}]\{x_n^{y_n}\} .Enter your answer in the form ,,,简答题

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BlackTom AI · Accepted Answer

We are given that the positive-value sequences {x_n} and {y_n} converge to 5 and 3, respectively. This means:
- lim x_n = 5 and lim y_n = 3.
- Since limits preserve simple operations under standard algebra, we can evaluate the limits of composed expressions by replacing x_n and y_n with their limits where appropriate, while also noting that limits of shifted sequences like y_{n+1} equal the limit of y_n if the limit exists.

First expression: x_n + y_{n+1}
- As n → ∞, x_n → 5 and y_{n+1} → lim y_n = 3.
- Therefore, x_n + y_{n+1} → 5 + 3 = 8.

Second expression: x_n y_{2n}
- Here x_n → 5 and y_{2n} → lim y_n = 3, since the subsequence y_{2n} converges to the same limit as y_n when the limit exists.
- Thus, x_n y_{2n} → 5 × 3 = 15.

Third expression: 2 x_n
- This is simply a scalar multiple of x_n, so 2 x_n → 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 of the power is 5^3.
- Therefore, x_n^{y_n} → 5^3 = 125.

Collecting the limits, the sequence limits are 8, 15, 10, 125 respectively.

If the positive-value sequences [math: {xn},{yn}]\{x_n\}, \{y_n\} have limits 5 and 3 respectively, what are the limits of the sequences [math: {xn+yn+1}]\{x_n+y_{n+1}\}, [math: {xny2n}]\{x_ny_{2n}\}, [math: {2xn}]\{2x_n\} and [math: {xnyn}]\{x_n^{y_n}\} .Enter your answer in the form ,,,简答题

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