Question

State whether the given series converges and explain why. $$ \sum_{n=1}^{\infty} \frac{1}{n+10^{80}} \text { (Hint: Rewrite using a change of index.) } $$

Short answer

The series diverges because it compares directly to the divergent harmonic series.

Step-by-step solution

Recognize the Series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{n+10^{80}} \). Our goal is to determine if this series converges.

Apply a Change of Index

To analyze the series, we perform a change of index by letting \( m = n + 10^{80} \) which implies that \( n = m - 10^{80} \). This gives us the new series \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \).

Compare with the Harmonic Series

Considering the series \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \), we can see it resembles the harmonic series which is \( \sum_{m=1}^{\infty} \frac{1}{m} \). The harmonic series is known to diverge.

Conclude Using a Comparison Test

Since \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \) is a sub-series of the harmonic series, and the harmonic series diverges, \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \) divergences as well. Consequently, the original series \( \sum_{n=1}^{\infty} \frac{1}{n+10^{80}} \) also diverges.

Key concepts

Harmonic Series

The harmonic series is a classic example in mathematical studies of series and sequences. It is expressed as \[ \sum_{n=1}^{\infty} \frac{1}{n} \] and is known to diverge, meaning that as more terms are added, the sum grows without bound. Despite the terms getting smaller, the infinite addition continuously increases the sum. This unique behavior distinguishes it from many other series.

• The harmonic series is vital in understanding series behavior.
• It often serves as a benchmark for determining the behavior of other, more complex series.

Recognizing patterns similar to the harmonic series in a given problem can provide insights into their convergence or divergence.

Divergence

Divergence specifies that a series grows indefinitely and does not approach a finite limit. For a series \( \sum_{n=1}^{\infty} a_n \) to diverge, the partial sums \( S_n = a_1 + a_2 + a_3 + \cdots + a_n \) should increase without reaching a steady value. In the context of the harmonic series, even though the terms become smaller, each subsequent sum still grows slowly but steadily.

• A diverging series never settles at a finite value.
• Understanding a series' divergence is crucial in differentiating it from convergent series.

Applying divergence can simplify analysis by eliminating the need for more complex calculations when examining series similar to the harmonic series.

Comparison Test

The Comparison Test is a helpful tool for analyzing the convergence or divergence of a series by relating it to a known benchmark series. If we have two series \( \sum a_n \) and \( \sum b_n \):

• If \( 0 \leq a_n \leq b_n \) for all \( n \), and \( \sum b_n \) converges, \( \sum a_n \) must also converge.
• If \( a_n \geq b_n \geq 0 \) for all \( n \), and \( \sum b_n \) diverges, then \( \sum a_n \) diverges as well.

In the given problem, \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \) is compared to the divergent harmonic series, \( \sum_{m=1}^{\infty} \frac{1}{m} \). As a sub-series of a divergent series, it also diverges by the Comparison Test.

Index Change

Index Change is a useful technique in the analysis of series, allowing for simplification and transformation into a more recognizable form. By changing the index in a series, we can often identify it as one that resembles a known form. For example, in the exercise, rewriting \( \sum_{n=1}^{\infty} \frac{1}{n+10^{80}} \) using \( m = n + 10^{80} \) turns it into \( \sum_{m=10^{80}+1}^{\infty} \frac{1}{m} \), which is easier to analyze.

• Re-indexing can uncover properties of series not immediately apparent.
• It helps bring into view similarities with well-studied series.

Index changes can be crucial in making connections between complex and simpler series, ultimately aiding in deciding convergence or divergence.