Infinite series appear throughout calculus, physics, engineering, and computer science because they let us represent complicated quantities as sums of simpler terms. A convergence test tells you whether an infinite sum approaches a finite value or grows without bound. Choosing the right test saves time and helps you avoid forcing a method that does not fit the series.
The key is to recognize the structure of the terms before doing long calculations.
A good strategy begins by checking whether the terms even approach zero, since a series cannot converge if lim a_n is not 0. After that, look for special forms such as geometric series, p-series, alternating signs, factorials, exponentials, or functions that are positive and decreasing. Ratio and root tests are powerful for factorials and nth powers, while comparison and integral tests work well for positive terms that resemble known benchmark series.
Alternating series require checking that the term sizes decrease to zero, not just that the signs alternate.
Key Facts
- Divergence test: if lim as n approaches infinity of a_n is not 0, then sum a_n diverges.
- Geometric series: sum ar^n converges if |r| < 1 and diverges if |r| >= 1.
- p-series: sum 1/n^p converges if p > 1 and diverges if p <= 1.
- Ratio test: let L = lim |a_(n+1)/a_n|. If L < 1 converge, if L > 1 diverge, if L = 1 inconclusive.
- Root test: let L = lim nth root of |a_n|. If L < 1 converge, if L > 1 diverge, if L = 1 inconclusive.
- Alternating series test: sum (-1)^n b_n converges if b_n decreases and lim b_n = 0.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, written as sum a_n.
- Convergence
- Convergence means the sequence of partial sums approaches a finite number.
- Divergence
- Divergence means the partial sums do not approach a finite number.
- Comparison test
- The comparison test determines convergence by comparing a positive-term series to a known larger or smaller benchmark series.
- Absolute convergence
- A series converges absolutely if the series of absolute values, sum |a_n|, converges.
Common Mistakes to Avoid
- Using the divergence test to prove convergence, which is wrong because lim a_n = 0 does not guarantee that sum a_n converges.
- Forgetting the absolute value in the ratio or root test, which is wrong because these tests measure the size of the terms and ignore sign changes.
- Applying the alternating series test without checking that b_n decreases, which is wrong because sign alternation and a zero limit are not enough by themselves.
- Choosing comparison series in the wrong direction, which is wrong because to prove convergence you need an upper bound by a convergent series, while to prove divergence you need a lower bound by a divergent series.
Practice Questions
- 1 Decide whether sum from n = 1 to infinity of 3(2/5)^n converges or diverges, and name the test you used.
- 2 Decide whether sum from n = 2 to infinity of n^2/(n^5 + 1) converges or diverges by comparing it to a p-series.
- 3 A series contains both n! and 4^n in its terms. Explain why the ratio test is likely a better first choice than the integral test.