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Chapter 11: Problem 10

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty}\left(n^{-1.4}+3 n^{-1.2}\right)$$

Short Answer

Expert verified
The series is convergent.

Step by step solution

01

Break down the series into simpler components

The given series is \( \sum_{n=1}^{\infty} \left(n^{-1.4} + 3n^{-1.2}\right) \). We can split this series into two separate series: \( \sum_{n=1}^{\infty} n^{-1.4} \) and \( 3\sum_{n=1}^{\infty} n^{-1.2} \). This allows us to analyze the convergence of each series individually.
02

Determine the convergence of each component using the p-series test

For the series \( \sum_{n=1}^{\infty} n^{-p} \), the series converges if \( p > 1 \) and diverges if \( p \leq 1 \). First, consider \( \sum_{n=1}^{\infty} n^{-1.4} \). Since 1.4 > 1, this series converges. Now, consider \( 3\sum_{n=1}^{\infty} n^{-1.2} \); since 1.2 > 1, it also converges based on the p-series test.
03

Use linearity to combine results

Since both \( \sum_{n=1}^{\infty} n^{-1.4} \) and \( 3\sum_{n=1}^{\infty} n^{-1.2} \) converge, we use the fact that the sum of two convergent series is also convergent. This means that the original series \( \sum_{n=1}^{\infty} \left(n^{-1.4} + 3n^{-1.2}\right) \) is convergent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

p-Series Test
Understanding the p-series test is crucial for analyzing series convergence. A p-series is in the form of \( \sum_{n=1}^{\infty} n^{-p} \). The test provides a simple criterion:
  • If \( p > 1 \), the series converges.
  • If \( p \leq 1 \), the series diverges.
This test is powerful because it allows us to easily determine convergence without complex calculations.
For our exercise, we applied the p-series test to the components \( n^{-1.4} \) and \( 3n^{-1.2} \). Given that both exponents (1.4 and 1.2) are greater than 1, the respective p-series converge.
Divergence
Divergence refers to a series not settling towards a finite limit. In simpler terms, if a series diverges, partial sums keep growing.
Understanding divergence is the counterpart to recognizing convergence. While the p-series test focuses on identifying convergence, it inherently tells us about divergence when \( p \leq 1 \).
In exercises like ours, when no divergence was present because \( p > 1 \) in both components, it's important to know divergence can signal when to stop looking for a finite sum. Knowing divergence helps in ruling out possibilities, streamlining our analysis.
Linearity in Series
The principle of linearity in series dictates that the summation of linear combinations of series behaves predictably. Essentially, we can add series term by term.
This property simplifies solving complex series problems. If two series are known to converge, their linear combination is also convergent.
In our example, both components \( \sum_{n=1}^{\infty} n^{-1.4} \) and \( 3\sum_{n=1}^{\infty} n^{-1.2} \) converged individually. Thus, combining them still results in convergence. Breaking complex series into manageable parts and leveraging linearity offers clearer paths to solutions.
Component Series Analysis
Component series analysis involves breaking a complicated series into simpler, manageable parts for detailed examination. This is akin to solving a puzzle by focusing on individual pieces first.
In our exercise, the original series \( \sum_{n=1}^{\infty} (n^{-1.4} + 3n^{-1.2}) \) was split into two distinct series.
  • First series: \( \sum_{n=1}^{\infty} n^{-1.4} \)
  • Second series: \( 3\sum_{n=1}^{\infty} n^{-1.2} \)
Through this analysis, you can individually apply convergence tests, making the problem much simpler. This strategy not only aids in clear understanding but significantly reduces the potential for errors.

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Most popular questions from this chapter

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