CalcRef

Topics · 18 formulas

Sequences & Series

Convergence tests, power series, Taylor and Maclaurin series.

Sequence Convergence

lim⁡n→∞an=L⇒{an} converges to L\lim_{n \to \infty} a_n = L \Rightarrow \{a_n\} \text{ converges to } L

A sequence {aₙ} converges if the limit of its terms as n→∞ exists and is finite.

Sequences

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Geometric Sequence

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

Each term is obtained by multiplying the previous term by a constant ratio r. Converges if |r| < 1.

Sequences

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Arithmetic Sequence

an=a1+(n−1)da_n = a_1 + (n-1)d

Each term is obtained by adding a constant difference d to the previous term. Always diverges (unless d = 0).

Sequences

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Series (Partial Sums)

∑n=1∞an=lim⁡N→∞∑n=1Nan=lim⁡N→∞SN\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} \sum_{n=1}^{N} a_n = \lim_{N \to \infty} S_N

An infinite series is the limit of its partial sums. If the limit exists and is finite, the series converges.

Series Types

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Geometric Series

∑n=0∞arn=a1−rif ∣r∣<1\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{if } |r| < 1

A geometric series converges if and only if |r| < 1, and its sum is a/(1-r).

Series Types

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p-Series

∑n=1∞1np converges if p>1, diverges if p≤1\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1 \text{, diverges if } p \leq 1

The p-series is a fundamental test case. The harmonic series (p = 1) diverges. For p > 1, the series converges.

Series Types

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Divergence Test

lim⁡n→∞an≠0⇒∑an diverges\lim_{n \to \infty} a_n \neq 0 \Rightarrow \sum a_n \text{ diverges}

If the terms do not approach zero, the series diverges. CAUTION: If the limit IS zero, the test is inconclusive (the series may still diverge).

Convergence Tests

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Integral Test

f(n)=an,  f positive, continuous, decreasing:∑an and ∫1∞f(x) dx both converge or both divergef(n) = a_n,\; f \text{ positive, continuous, decreasing}: \sum a_n \text{ and } \int_1^{\infty} f(x)\, dx \text{ both converge or both diverge}

If f is positive, continuous, and decreasing for x ≥ N, then the series and the improper integral either both converge or both diverge.

Convergence Tests

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Comparison Test

0≤an≤bn:∑bn conv.⇒∑an conv.;∑an div.⇒∑bn div.0 \leq a_n \leq b_n: \sum b_n \text{ conv.} \Rightarrow \sum a_n \text{ conv.}; \quad \sum a_n \text{ div.} \Rightarrow \sum b_n \text{ div.}

Compare with a known series. If the larger series converges, so does the smaller. If the smaller diverges, so does the larger.

Convergence Tests

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Limit Comparison Test

lim⁡n→∞anbn=L>0⇒∑an and ∑bn both converge or both diverge\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0 \Rightarrow \sum a_n \text{ and } \sum b_n \text{ both converge or both diverge}

If the ratio of terms approaches a positive finite limit, the two series have the same convergence behavior.

Convergence Tests

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Alternating Series Test

∑(−1)nbn converges if bn>0,  bn+1≤bn,  lim⁡bn=0\sum (-1)^n b_n \text{ converges if } b_n > 0,\; b_{n+1} \leq b_n,\; \lim b_n = 0

An alternating series converges if the absolute values of terms are decreasing and approach zero. The error after N terms is at most |b_{N+1}|.

Convergence Tests

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Ratio Test

L=lim⁡n→∞∣an+1an∣:L<1⇒conv.,  L>1⇒div.,  L=1⇒inconclusiveL = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|: \quad L < 1 \Rightarrow \text{conv.},\; L > 1 \Rightarrow \text{div.},\; L = 1 \Rightarrow \text{inconclusive}

The ratio test: compute the limit of the absolute ratio of consecutive terms. Particularly useful for series with factorials or exponentials.

Convergence Tests

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Power Series

∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n (x - a)^n

A power series centered at a. It converges for |x - a| < R (radius of convergence) and diverges for |x - a| > R. Check endpoints separately.

Power & Taylor Series

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Radius of Convergence

R=lim⁡n→∞∣cncn+1∣orR=1lim sup⁡∣cn∣1/nR = \lim_{n \to \infty} \left|\frac{c_n}{c_{n+1}}\right| \quad \text{or} \quad R = \frac{1}{\limsup |c_n|^{1/n}}

The radius of convergence R determines where a power series converges absolutely. Often found via the ratio or root test.

Power & Taylor Series

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Taylor Series

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

The Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point a.

Power & Taylor Series

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Maclaurin Series

f(x)=∑n=0∞f(n)(0)n!xn(Taylor series at a=0)f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n \quad \text{(Taylor series at } a = 0\text{)}

A Maclaurin series is a Taylor series centered at a = 0. Common Maclaurin series: eˣ, sin x, cos x, 1/(1-x), ln(1+x).

Power & Taylor Series

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Taylor Polynomial Error Bound

∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \leq \frac{M}{(n+1)!}|x - a|^{n+1}

The Lagrange error bound (Taylor remainder): the error in the nth-degree Taylor approximation is bounded by M|x-a|^(n+1)/(n+1)!, where M is the max of |f⁽ⁿ⁺¹⁾| on the interval.

Power & Taylor Series

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Root Test

L=lim⁡n→∞∣an∣n:L<1⇒conv.,  L>1⇒div.,  L=1⇒inconclusiveL = \lim_{n \to \infty} \sqrt[n]{|a_n|}: \quad L < 1 \Rightarrow \text{conv.},\; L > 1 \Rightarrow \text{div.},\; L = 1 \Rightarrow \text{inconclusive}

The root test: compute the limit of the nth root of |aₙ|. Useful when terms involve nth powers.

Convergence Tests

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