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.

MTH1030 -1035 - S1 2025 MTH1030/5 Week 8 lesson quiz: Infinite sequences and series

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 ,,,

View Explanation

Log in for full answers

Similar Questions

Let \(f(x) = x^2 \sin(x)\) and let \(x_n=1/n\). What is the limit of the sequence \(\{f(x_n)\}\)?

Let [math: f(x)=x2sin⁡(x)]f(x) = x^2 \sin(x) and let [math: xn=1/n]x_n=1/n. What is the limit of the sequence [math: {f(xn)}]\{f(x_n)\}?

Let [math: f(x)=x2]f(x) = x^2 and [math: xn=2+1/n]x_n=2+1/n.What is the limit of the sequence [math: {f(xn)}]\{f(x_n)\}?

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 ,,,

Let { 𝑎 𝑛 } 𝑛 = 0 ∞ be a sequence. Which statements are equivalent to " { 𝑎 𝑛 } 𝑛 = 0 ∞ is divergent to ∞ " ? Select all the correct answers.

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler