calculate the first three partial sums of the series.

\\sum_{n=1}^{\infty} \frac{n}{n+5}\

$s_1 =$

$s_2 =$

$s_3 =$

use the sequence of partial sums to determine whether the series converges or diverges.

* converges
* diverges

Question

calculate the first three partial sums of the series.

\\sum_{n=1}^{\infty} \frac{n}{n+5}\

$s_1 =$

$s_2 =$

$s_3 =$

use the sequence of partial sums to determine whether the series converges or diverges.

* converges
* diverges

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Partial Sums", "Series Convergence" ], "new_concepts": [], "current_concepts": [ "Partial Sums", "Series Convergence", "Sigma Notation" ] } ## Calculate the terms of the series $$ \begin{aligned} a_n &= \frac{n}{n+5} \ a_1 &= \frac{1}{1+5} = \frac{1}{6} \ a_2 &= \frac{2}{2+5} = \frac{2}{7} \ a_3 &= \frac{3}{3+5} = \frac{3}{8} \end{aligned} $$ ## Compute the first three partial sums $$ \begin{aligned} S_1 &= a_1 = \frac{1}{6} \ S_2 &= S_1 + a_2 = \frac{1}{6} + \frac{2}{7} = \frac{7 + 12}{42} = \frac{19}{42} \ S_3 &= S_2 + a_3 = \frac{19}{42} + \frac{3}{8} = \frac{19 \times 4 + 3 \times 21}{168} = \frac{76 + 63}{168} = \frac{139}{168} \end{aligned} $$ ## Determine convergence of the series $$ \begin{aligned} \lim_{n \to \infty} a_n &= \lim_{n \to \infty} \frac{n}{n+5} = 1 eq 0 \ \lim_{n \to \infty} S_n &= \infty \implies \text{diverges} \end{aligned} $$ ## Question 1 $S_1 =$ $\frac{1}{6}$ ## Question 2 $S_2 =$ $\frac{19}{42}$ ## Question 3 $S_3 =$ $\frac{139}{168}$ ## Question 4 converges

diverges

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Infinite Series Representation" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the terms of the series

Compute the first three partial sums

Determine convergence of the series

Answer:

Question 1

Question 2

Question 3

Question 4