The divergence test and integral test are powerful tools for analyzing infinite series. They help determine if a series converges or diverges, which is crucial for understanding its behavior and potential applications.

These tests, along with comparison methods and telescoping series, form a toolkit for tackling various types of series. By mastering these techniques, you'll be able to evaluate complex sums and make informed decisions about their convergence or divergence.

The Divergence Test

Divergence test for series

• Determines if a series diverges by examining the limit of its terms
  • If $\lim_{n \to \infty} a_n \neq 0$ or the limit does not exist, the series $\sum_{n=1}^{\infty} a_n$ diverges ($\sum_{n=1}^{\infty} \frac{1}{n}$, harmonic series)
  • If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive and the series may converge or diverge ($\sum_{n=1}^{\infty} \frac{1}{n^2}$, $p$-series with $p=2$)
• Useful for identifying divergent series quickly without calculating the sum
• Does not provide information about convergence

The Integral Test

Integral test for convergence

• Determines the convergence or divergence of a series by comparing it to an improper integral
• For a series $\sum_{n=1}^{\infty} a_n$, let $f(x)$ be a continuous, positive, and decreasing function on $[1, \infty)$ such that $f(n) = a_n$ for all $n \geq 1$
  • If $\int_1^{\infty} f(x) dx$ converges, then $\sum_{n=1}^{\infty} a_n$ converges ($\sum_{n=1}^{\infty} \frac{1}{n^2}$ and $\int_1^{\infty} \frac{1}{x^2} dx$)
  • If $\int_1^{\infty} f(x) dx$ diverges, then $\sum_{n=1}^{\infty} a_n$ diverges ($\sum_{n=1}^{\infty} \frac{1}{n}$ and $\int_1^{\infty} \frac{1}{x} dx$)
• Useful for series involving logarithms, exponentials, or powers ($\sum_{n=1}^{\infty} \frac{1}{n \ln n}$, $\sum_{n=1}^{\infty} e^{-n}$)
• Requires the function to be monotonic (decreasing) for the test to be valid

Remainder terms for series estimation

• The remainder term $R_n$ is the difference between the series value $S$ and the $n$-th partial sum $S_n$
  • $R_n = S - S_n$, represents the error when approximating the series value with a partial sum
• For a convergent series $\sum_{n=1}^{\infty} a_n$ with positive, decreasing, and continuous terms $a_n$ for $n \geq 1$:
  1. $\int_{n+1}^{\infty} a_n dn \leq R_n \leq \int_n^{\infty} a_n dn$
  2. These integrals provide lower and upper bounds for $R_n$
• Calculating remainder term bounds helps:
  • Estimate the series value ($\sum_{n=1}^{\infty} \frac{1}{n^2}$, approximate value to within $0.01$)
  • Determine the number of terms needed for a desired accuracy ($\sum_{n=1}^{\infty} e^{-n}$, how many terms for an error less than $0.001$?)

Additional Convergence Tests

Comparison and Limit Comparison Tests

• Compares the given series to a known convergent or divergent series
• If $0 \leq a_n \leq b_n$ for all $n \geq N$:
  • If $\sum b_n$ converges, then $\sum a_n$ converges
  • If $\sum a_n$ diverges, then $\sum b_n$ diverges
• Limit comparison test uses the limit of the ratio of terms
• Useful for series that are difficult to evaluate directly

Telescoping Series

• A series where terms cancel out, leaving only a finite number of terms
• Often results in a simple expression for the sum
• Example: $\sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1})$