Sequences & Series

🔢 Sequences & Series

Infinite sequences and series allow us to approximate complex functions using simple polynomials. This is the foundation of many numerical algorithms.

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🟢 1. Sequences

A sequence is an ordered list of numbers $\{a_n\}$.

• Convergence: A sequence $\{a_n\}$ converges to $L$ if $\lim_{n \to \infty} a_n = L$.
• Divergence: If the limit doesn’t exist or is infinite.

Common Sequences

• Arithmetic: $a_n = a_1 + (n-1)d$.
• Geometric: $a_n = a_1 r^{n-1}$ (converges if $|r| < 1$ to $0$).

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🟡 2. Infinite Series

An infinite series is the sum of a sequence: $\sum_{n=1}^\infty a_n$.

Geometric Series

$\sum_{n=0}^\infty ar^n$ converges if and only if $|r| < 1$. $\text{Sum} = \frac{a}{1 - r}$

Divergence Test

If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ must diverge.

Caution

The converse is not true! For example, the harmonic series $\sum \frac{1}{n}$ has terms that go to zero, but the series still diverges.

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🔴 3. Tests for Convergence

To determine if a series converges without finding its sum:

1. Integral Test: Compare the series to $\int_1^\infty f(x) dx$.
2. P-Series Test: $\sum \frac{1}{n^p}$ converges if $p > 1$.
3. Comparison Test: Compare with a known convergent/divergent series.
4. Ratio Test: $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$. Converges if $L < 1$, diverges if $L > 1$.
5. Alternating Series Test: For sums like $\sum (-1)^n b_n$.

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🎯 4. Power Series and Taylor Series

A Power Series is a function of the form $f(x) = \sum_{n=0}^\infty c_n (x - a)^n$.

Taylor and Maclaurin Series

A Taylor Series represents a function $f(x)$ as an infinite sum of its derivatives at a point $a$: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n$

If $a = 0$, it is called a Maclaurin Series: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots$

Common Maclaurin Series

• $e^x = \sum \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$
• $\sin(x) = \sum (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \dots$
• $\cos(x) = \sum (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \dots$

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💡 Practical Example: Calculating Pi with a Series

The Gregory-Leibniz series for $\pi/4$: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$

def estimate_pi(n_terms):
    pi_over_4 = 0
    for n in range(n_terms):
        pi_over_4 += ((-1)**n) / (2*n + 1)
    return pi_over_4 * 4

# More terms = more precision
print(f"PI (10 terms): {estimate_pi(10)}")
print(f"PI (10,000 terms): {estimate_pi(10000)}")

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🚀 Key Takeaways

• Sequences are lists, series are sums.
• Convergence tests tell us if an infinite sum is finite.
• Taylor Series allow us to approximate any “well-behaved” function with a polynomial.