In mathematics, particularly in calculus and series analysis, convergence and divergence describe the behavior of sequences and series as they progress toward infinity.

1. Convergence:

A sequence or series converges if it approaches a specific value (finite number) as the number of terms increases.

For a sequence:

A sequence $\{a_n\}$ converges to $L$ if: $\lim_{n \to \infty} a_n = L$ For example, the sequence $a_n = \frac{1}{n}$ converges to 0 because as $n \to \infty$, the terms get arbitrarily close to 0.

For a series:

A series $\sum_{n=1}^\infty a_n$ converges if the partial sums $\{S_n\}$, defined as: $S_n = \sum_{k=1}^n a_k$ approach a finite value $S$ as $n \to \infty$: $\lim_{n \to \infty} S_n = S$ For example, the geometric series $\sum_{n=0}^\infty r^n$ converges to $\frac{1}{1-r}$ if $|r| < 1$.

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2. Divergence:

A sequence or series diverges if it does not converge, meaning it does not approach a finite limit.

For a sequence:

If the sequence $\{a_n\}$ does not settle to a single finite value, it diverges. Examples include:

• The sequence $a_n = n$ diverges to infinity.
• The sequence $a_n = (-1)^n$ oscillates and does not converge to any single value.

For a series:

A series $\sum_{n=1}^\infty a_n$ diverges if its partial sums do not approach a finite limit. For example:

• The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges even though the terms $\frac{1}{n} \to 0$.
• Any series $\sum_{n=1}^\infty a_n$ where $\lim_{n \to \infty} a_n \neq 0$ must diverge (Divergence Test).

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3. Tests for Convergence/Divergence:

Here are some common tests to analyze convergence or divergence:

a. Divergence Test:

If $\lim_{n \to \infty} a_n \neq 0$, the series $\sum a_n$ diverges. However, $\lim_{n \to \infty} a_n = 0$ does not guarantee convergence.

b. Ratio Test:

For $\sum a_n$, compute $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$:

• If $L < 1$, the series converges absolutely.
• If $L > 1$ or $L = \infty$, the series diverges.
• If $L = 1$, the test is inconclusive.

c. Comparison Test:

If $0 \leq a_n \leq b_n$ for all $n$:

• If $\sum b_n$ converges, then $\sum a_n$ converges.
• If $\sum a_n$ diverges, then $\sum b_n$ diverges.

d. Integral Test:

For a positive, decreasing function $f(n)$ corresponding to the terms $a_n$:

• If $\int_1^\infty f(x)\,dx$ converges, then $\sum a_n$ converges.
• If $\int_1^\infty f(x)\,dx$ diverges, then $\sum a_n$ diverges.

e. Alternating Series Test:

For an alternating series $\sum (-1)^n b_n$ where $b_n > 0$:

• If $b_n \to 0$ and $b_n$ decreases, the series converges.

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Questions You May Have:

1. Can you explain how to determine if a sequence converges or diverges?
2. What is the difference between absolute and conditional convergence?
3. How does the integral test work, and when should it be applied?
4. Are there examples of series that are neither geometric nor harmonic but converge?
5. Why does the harmonic series diverge even though its terms approach zero?

Tip:

When analyzing a series, always check the behavior of individual terms ($a_n$) as $n \to \infty$. If the terms don’t approach zero, the series cannot converge!