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": 2, "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" ] } ## Calculate the second partial sum $S_2$ $$ \begin{aligned} a_n &= \frac{n}{n+5} \ S_2 &= a_1 + a_2 = \frac{1}{1+5} + \frac{2}{2+5} \ S_2 &= \frac{1}{6} + \frac{2}{7} = \frac{7 + 12}{42} = \frac{19}{42} \end{aligned} $$ ## Determine the 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 $S_2 =$ $\frac{19}{42}$ ## Question 2 converges

diverges

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Partial Sums" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the second partial sum \(S_2\)

Determine the convergence of the series

Answer:

Question 1

Question 2