Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n}{2+n 5^{n}} $$

Step-by-Step Solution

Answer

The series converges by the ratio test.

Step 1: Simplify the General Term

First, let's simplify the general term for the series: \( \frac{n}{2+n 5^n} \). The denominator grows very quickly because it includes \( 5^n \), which is an exponential term. For large values of \( n \), \( n 5^n \) dominates over the constant 2.

Step 2: Apply the Ratio Test (asymptotically)

Consider using the ratio test to determine the convergence of the series. The ratio test involves finding the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|,\]where \( a_n = \frac{n}{2+n 5^n} \). Calculate the ratio:\[\frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{2+(n+1)5^{n+1}}}{\frac{n}{2+n5^n}}.\]Simplifying this involves a complex setup but noting the dominance of \( 5^n \), notice that asymptotically:\[ \frac{n5^n}{(n+1)5^{n+1}} \approx \frac{1}{5}\]This implies \( L = \frac{1}{5} < 1 \), suggesting the series converges by the ratio test.

Step 3: Conclusion

Based on the result from the ratio test, we have \( L < 1 \), which tells us that the series \( \sum_{n=1}^{\infty} \frac{n}{2+n 5^n} \) converges.

Key Concepts

Ratio Test

The Ratio Test is a popular method to determine the convergence or divergence of an infinite series. If you have a series in the form of \( \sum a_n \), the Ratio Test can tell us whether it converges absolutely, diverges, or the test is inconclusive.

• Start by considering the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the Ratio Test is inconclusive, and you’ll need another test to make a conclusion.

In the exercise, by using the Ratio Test, the calculated limit for the series \( \sum \frac{n}{2+n 5^n} \) was found to be \( \frac{1}{5} \), which is less than 1. Hence, the series converges. This result effectively tells us that as \( n \) becomes very large, the terms of the series become smaller at a faster than linear pace, leading to convergence.

Exponential Growth

Exponential growth is a powerful mathematical concept that describes processes increasing over time, wherein the growth rate is proportional to the value of the function.
For example, in this series, the term \( 5^n \) contributes to exponential growth by rapidly increasing as \( n \) increases. Here are some characteristics:

• Exponential terms usually overpower linear or polynomial terms, especially as \( n \) becomes very large.
• In the context of the series \( \sum \frac{n}{2+n 5^n} \), the rapid growth of \( 5^n \) makes its influence dominant in the denominator, ensuring that the fraction as a whole shrinks toward zero as n increases.

Understanding exponential growth is crucial for recognizing why certain series converge or diverge. When such growth is present, it’s often the case that exponential growth in the denominator will result in the sequence of terms of the series diminishing quickly to zero, thus supporting convergence.

Series Simplification

Simplifying a series is a necessary step when you first approach problems related to series convergence or divergence. It often involves identifying and focusing on the dominant terms that heavily influence the behavior of the series.
In the series \( \frac{n}{2 + n5^n} \), the term \( 5^n \) is the most significant contributor in the denominator, especially for large \( n \). Here are some considerations for simplification:

• Ignore lower order terms when they don’t significantly impact the behavior of the series at large \( n \). In this case, "2" becomes negligible compared to \( n5^n \).
• Focus on terms that dictate the growth or decline speed. For example, since \( 5^n \) grows extremely fast, it confirms the decrease in terms like \( \frac{n}{5^n} \) to near zero, promoting convergence.

By simplifying, you can better apply tests like the Ratio Test, and gain insights into the series’ behavior, making it easier to deduce the ultimate convergence or divergence outcome.