Question

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. $$\sum_{n=0}^{\infty} \sqrt{n}$$

Name: Problem 19, Chapter 30: Basic Technical Mathematics with Calculus
Uploaded: 2021-06-09T01:11:12-08:00
Duration: 1 min 10 s
Channel: Tyler Moulton
Description: Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. ∑n=0^∞ √(n)

   Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=0}^{\infty} \sqrt{n}$$

Basic Technical Mathematics with Calculus

Allyn J. Washington,… 11th Edition

Chapter 30, Problem 19

Instant Answer

Solved by Expert Tyler Moulton

06/09/2021

Step 1

The partial sum is the sum of the terms of the series up to a certain point. The first term of the series is $\sqrt{0}=0$, so the first partial sum $S_1=0$. Show more…

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Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. $$\sum_{n=0}^{\infty} \sqrt{n}$$

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Key Concepts

Partial Sums

Partial sums are the sums of the first n terms of a series. They are used to approximate the series and analyze its convergence or divergence by observing how these sums behave as n increases.

Convergence and Divergence

A series is convergent if its sequence of partial sums approaches a finite limit; if not, the series is divergent. This is a critical concept when determining the long-term behavior of an infinite series.

Infinite Series

An infinite series is the sum of infinitely many terms, and its convergence is determined by whether the sequence of partial sums approaches a finite limit. Studying these sums helps in understanding the overall behavior of the series.

Divergence Test

The divergence test, or nth-term test, states that if the limit of the terms of a series does not tend to zero, then the series diverges. Since in this series the term sqrt(n) grows without bound and does not approach zero, the series diverges.

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