Training Sequences & Series Infinite Series & Convergence
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Infinite Series & Convergence

26 min Sequences & Series
An infinite series Σ aₙ converges if the sequence of partial sums Sₙ approaches a finite limit. The geometric series Σ arⁿ converges to a/(1 − r) when |r| < 1 and diverges otherwise. Convergence tests include the nth-term test (if aₙ ↛ 0, the series diverges), the comparison test, the ratio test (|aₙ₊₁/aₙ| → L: converges if L < 1), the root test, and the integral test. Alternating series Σ(−1)ⁿ bₙ converge if bₙ is decreasing and bₙ → 0 (Leibniz criterion). Absolute convergence (Σ|aₙ| converges) implies convergence, but conditional convergence (converges but not absolutely) requires more care.

Infinite Series & Convergence

Infinite Series

$$\sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} S_n$$

If the limit exists and is finite, the series converges; otherwise it diverges.

Infinite Geometric Series

If $|r| < 1$:

$$\sum_{k=0}^{\infty} a_1 r^k = \frac{a_1}{1 - r}$$

If $|r| \geq 1$, the series diverges.

Divergence Test

If $\lim_{n\to\infty} a_n \neq 0$, then $\sum a_n$ diverges.

Warning: If $\lim a_n = 0$, the series may still diverge (e.g., harmonic series).

Telescoping Series

A series where consecutive terms cancel, leaving only the first and last pieces. Write partial fractions and cancel.

Example 1

Find the sum: $\sum_{k=0}^{\infty} \left(\frac{1}{2}\right)^k$.

  1. $a_1 = 1$, $r = 1/2$.
  2. Since $|r| < 1$:
  3. $$S = \frac{1}{1 - 1/2} = 2$$
Example 2

Does $\sum_{k=1}^{\infty} \frac{k}{k+1}$ converge?

  1. $\lim_{k\to\infty} \frac{k}{k+1} = 1 \neq 0$.
  2. By the Divergence Test, the series diverges.
Example 3

Write the repeating decimal $0.333\ldots$ as a fraction.

  1. $0.333\ldots = \sum_{k=1}^{\infty} 3 \cdot (0.1)^k = \frac{0.3}{1 - 0.1} = \frac{0.3}{0.9} = \frac{1}{3}$

Practice Problems

1. $\sum_{k=0}^{\infty} \left(\frac{1}{3}\right)^k$.
2. $\sum_{k=0}^{\infty} \left(\frac{3}{4}\right)^k$.
3. Does $\sum_{k=1}^{\infty} \frac{1}{k}$ converge?
4. Apply the divergence test: $\sum (-1)^k$.
5. $\sum_{k=1}^{\infty} 5 \cdot (0.2)^k$.
6. Write $0.\overline{6}$ as a fraction using geometric series.
7. Does $\sum_{k=0}^{\infty} 2^k$ converge?
8. $\sum_{k=1}^{\infty} \frac{1}{k(k+1)}$ (telescoping).
9. Geometric: $a_1 = 10$, $r = -0.5$. Sum?
10. $\sum_{k=1}^{\infty} \frac{3}{10^k}$.
11. What is the divergence test?
12. If $|r| = 1$, does the geometric series converge?
Show Answer Key

1. $\frac{1}{1-1/3} = 3/2$

2. $\frac{1}{1-3/4} = 4$

3. No — it is the harmonic series (diverges)

4. $\lim (-1)^k$ does not exist (alternates), so diverges

5. $\frac{5 \cdot 0.2}{1 - 0.2} = \frac{1}{0.8} = 1.25$

6. $\frac{0.6}{1-0.1} = \frac{0.6}{0.9} = 2/3$

7. No, $|r| = 2 \geq 1$

8. Partial fractions: $\frac{1}{k} - \frac{1}{k+1}$. Telescopes to $1$.

9. $\frac{10}{1-(-0.5)} = \frac{10}{1.5} = 20/3$

10. $\frac{3/10}{1-1/10} = \frac{3/10}{9/10} = 1/3$

11. If $\lim a_n \neq 0$, the series diverges

12. No, it diverges

∞ Geometric Series Convergence
S∞ = a/(1−r)
Sₙ (partial sum)
Error |S∞ − Sₙ|
Summation Notation & Partial Sums Power Series & Taylor Series