Can you add infinitely many numbers and get a finite answer? Sometimes yes, sometimes no, and deciding which is the job of convergence tests. This guide walks you through the essential tests used in university calculus and A Level Further Maths, shows when to use each one, and works through examples step by step.
Convergence and Divergence
An infinite series is a sum
$$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots $$
We look at the partial sums \( S_N = a_1 + a_2 + \cdots + a_N \). If \( S_N \) approaches a finite number as \( N \to \infty \), the series converges. Otherwise it diverges.
The Essential Tests
1. The \( n \)th Term (Divergence) Test
If \( \lim_{n \to \infty} a_n \neq 0 \), the series diverges.
Warning: if the limit is 0, this test tells you nothing. The harmonic series \( \sum \frac{1}{n} \) has terms tending to 0 but still diverges.
2. The Geometric Series Test
A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if \( |r| < 1 \), with sum:
$$ \sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} $$
It diverges if \( |r| \ge 1 \).
3. The \( p \)-Series Test
$$ \sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1 \text{ and diverges if } p \le 1 $$
So \( \sum \frac{1}{n^2} \) converges, while \( \sum \frac{1}{n} \) and \( \sum \frac{1}{\sqrt{n}} \) diverge.
4. The Comparison Test
For series with positive terms:
- If \( 0 \le a_n \le b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
- If \( a_n \ge b_n \ge 0 \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.
5. The Limit Comparison Test
If \( a_n, b_n > 0 \) and
$$ \lim_{n \to \infty} \frac{a_n}{b_n} = L, \qquad 0 < L < \infty, $$
then both series converge or both diverge. This is often easier than the direct comparison test.
6. The Ratio Test
Let
$$ L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| $$
- \( L < 1 \): converges (absolutely).
- \( L > 1 \): diverges.
- \( L = 1 \): inconclusive.
The ratio test is ideal for series with factorials or exponentials like \( 2^n \).
7. The Alternating Series Test
A series \( \sum (-1)^{n+1} b_n \) with \( b_n > 0 \) converges if:
- \( b_n \) is decreasing, and
- \( \lim_{n \to \infty} b_n = 0 \).
Other tests, such as the root test and integral test, are also useful, but the seven above handle most exam questions.
Worked Example 1: The \( n \)th Term Test
Does \( \displaystyle\sum_{n=1}^{\infty} \frac{n}{2n + 1} \) converge?
Step 1: Find the limit of the terms. Divide top and bottom by \( n \):
$$ \lim_{n \to \infty} \frac{n}{2n + 1} = \lim_{n \to \infty} \frac{1}{2 + \frac{1}{n}} = \frac{1}{2} $$
Step 2: Conclude. Since \( \frac{1}{2} \neq 0 \), the series diverges.
Worked Example 2: A Geometric Series
Find the sum of \( \displaystyle\sum_{n=0}^{\infty} 3\left(\frac{2}{3}\right)^n \).
Step 1: Identify \( a \) and \( r \). \( a = 3 \), \( r = \frac{2}{3} \).
Step 2: Check convergence. \( |r| = \frac{2}{3} < 1 \), so it converges.
Step 3: Find the sum.
$$ \frac{a}{1 - r} = \frac{3}{1 - \frac{2}{3}} = \frac{3}{\frac{1}{3}} = 9 $$
Worked Example 3: The Ratio Test
Does \( \displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n} \) converge?
Step 1: Form the ratio.
$$ \frac{a_{n+1}}{a_n} = \frac{(n + 1)^2}{2^{n+1}} \cdot \frac{2^n}{n^2} = \frac{(n + 1)^2}{2n^2} $$
Step 2: Take the limit.
$$ L = \lim_{n \to \infty} \frac{1}{2}\left(\frac{n + 1}{n}\right)^2 = \frac{1}{2}(1)^2 = \frac{1}{2} $$
Step 3: Conclude. \( L = \frac{1}{2} < 1 \), so the series converges.
Worked Example 4: Comparison Tests
(a) Does \( \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 + 3} \) converge?
For every \( n \ge 1 \), \( n^2 + 3 > n^2 \), so
$$ 0 < \frac{1}{n^2 + 3} < \frac{1}{n^2} $$
Since \( \sum \frac{1}{n^2} \) is a convergent \( p \)-series (\( p = 2 > 1 \)), the given series converges by the comparison test.
(b) Does \( \displaystyle\sum_{n=1}^{\infty} \frac{2n + 1}{n^3 + n} \) converge?
For large \( n \), the terms behave like \( \frac{2n}{n^3} = \frac{2}{n^2} \), so compare with \( b_n = \frac{1}{n^2} \):
$$ \lim_{n \to \infty} \frac{2n + 1}{n^3 + n} \cdot \frac{n^2}{1} = \lim_{n \to \infty} \frac{2n^3 + n^2}{n^3 + n} = 2 $$
Since \( 0 < 2 < \infty \) and \( \sum \frac{1}{n^2} \) converges, the series converges by the limit comparison test.
Worked Example 5: The Alternating Harmonic Series
Does \( \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \cdots \) converge?
Here \( b_n = \frac{1}{n} \). It is positive, decreasing, and \( \lim b_n = 0 \). So the series converges by the alternating series test.
However, \( \sum \frac{1}{n} \) (without the signs) diverges, so this series is conditionally convergent, not absolutely convergent.
Tips: Which Test Should I Use?
- Always try the \( n \)th term test first. If the terms do not go to 0, you are done.
- Geometric or \( p \)-series form? Use those tests directly.
- Factorials or powers like \( 2^n \)? Use the ratio test.
- Rational function of \( n \)? Use limit comparison with \( \frac{1}{n^p} \), where \( p \) is the difference of the highest powers.
- Alternating signs? Use the alternating series test, then check absolute convergence.
Common Mistakes
- Concluding convergence because \( a_n \to 0 \). This is never enough on its own.
- Using the ratio test on \( p \)-series. It always gives \( L = 1 \), which is inconclusive.
- Comparing in the wrong direction, such as showing a series is smaller than a divergent one, which proves nothing.
- Forgetting absolute values in the ratio test for series with negative terms.
Key Takeaways
- A series converges when its partial sums approach a finite limit.
- If the terms do not tend to 0, the series diverges, but terms tending to 0 do not guarantee convergence.
- Geometric series converge for \( |r| < 1 \); \( p \)-series converge for \( p > 1 \).
- Use ratio tests for factorials and exponentials, comparison tests for rational expressions, and the alternating series test for alternating signs.
Need help choosing the right test or proving convergence in your assignment? Amathify's subject experts and PhD scholars provide detailed step-by-step solutions. Message us on WhatsApp or use the "Ask a Question" form.
Stuck on a problem or assignment?
Our experts and PhD scholars solve maths, physics and chemistry — step by step.