Question

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. $$ \left.\sum_{n=1}^{\infty} \frac{1}{2 n+1} \text { (Hint: Follow the reasoning for } \sum_{n=1}^{\infty} \frac{1}{n} .\right) $$

Short answer

The series diverges, similar to the harmonic series.

Step-by-step solution

Define Partial Sums

A partial sum, denoted as \( S_N \), is the sum of the first \( N \) terms of a series. For the series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \), the partial sum \( S_N \) is given by \( S_N = \sum_{n=1}^{N} \frac{1}{2n+1} \).

Compare with the Harmonic Series

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \), known as the harmonic series, is known to diverge. Since for all \( n \), \( \frac{1}{2n+1} > \frac{1}{2n+2} \), our series terms are similar in form and behavior to the harmonic series in terms of growth.

Consider the Divergence Test

The Divergence Test states that if \( \lim_{n \to \infty} a_n eq 0 \), the series \( \sum a_n \) diverges. In our series, \( a_n = \frac{1}{2n+1} \), and the limit as \( n \to \infty \) is 0. However, this test is inconclusive since the limit is 0. Further analysis is needed.

Analyze the Growth of Partial Sums

Consider \( T_N = \sum_{n=1}^{N} \frac{1}{2n+1} \). Note that \( \frac{1}{2n+1} \), like \( \frac{1}{n} \), does not decrease to 0 faster than \( \frac{1}{n} \). The sum \( T_N \) will not approach a finite limit, indicating divergence.

Conclusion

Since the terms of \( \frac{1}{2n+1} \) resemble those of the harmonic series \( \frac{1}{n} \) and the respective partial sums \( S_N \) tend to increase indefinitely, the series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \) diverges.

Key concepts

Partial Sums

In the context of calculus and series, partial sums can provide insight into whether a series converges or diverges. A partial sum is essentially the total sum of the first few terms of a sequence. When dealing with an infinite series, we denote the partial sum of the first \( N \) terms as \( S_N \). In essence, if the sequence of partial sums \( \{ S_N \} \) tends to a fixed number as \( N \) becomes very large, the series converges. Conversely, if \( \{ S_N \} \) grows indefinitely, the series diverges.
For the series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \), the partial sum \( S_N = \sum_{n=1}^{N} \frac{1}{2n+1} \) does not settle at a particular value, hence indicating divergence. Understanding partial sums helps you see the overall behavior of a series.

Divergence Test

The Divergence Test is one of the first tools you can use to ascertain whether an infinite series diverges. This test states that if the limit of the sequence terms \( a_n \) does not equal zero as \( n \) approaches infinity, then the series \( \sum a_n \) must diverge. While the divergence test is a straightforward method, it doesn’t always conclude the matter.
For instance, when the limit of \( a_n = \frac{1}{2n+1} \) approaches 0 as \( n \) grows larger, this test cannot determine convergence or divergence; this is where the test becomes inconclusive. Therefore, even though we apply it at the start, additional analysis is usually needed when it is inconclusive.

Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a famous example of a divergent series. Despite its terms approaching 0 as \( n \) grows, the series as a whole does not converge to any finite value. This characteristic makes the harmonic series particularly interesting in studies of series and sequences.
In comparing any series to the harmonic series, you often look for a similar rate of term decrease and behavior. When a series like \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \) has terms that behave similarly to \( \frac{1}{n} \), it often indicates divergence. Understanding the harmonic series helps contextualize why some series diverge even though individual terms shrink towards zero.

Series Divergence

Series divergence occurs when the sequence of partial sums does not head towards a finite limit. While many factors can unveil divergence, such as similarities to known divergent series such as the harmonic series, using a combination of approaches provides a stronger understanding.
The series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \) diverges because its partial sums continue to grow without bound. This perpetual growth reaffirms that the sum doesn’t settle onto a number, thus reiterating divergence.

• Compare to known divergent series.
• Use the divergence test if applicable.
• Observe the partial sums for unlimited increase.

Equipped with these methods, you can better tackle the diverse array of series in calculus.