A p-series is $$\sum_{n=1}^{\infty}\frac{1}{n^p}=\frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+...$$ converges if p > 1, diverges if p ≤ 1. 10.5

$$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+...$$ always DIVERGES 10.5

Comparison Test for Convergence

Compare to a known series: If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, then ∑aₙ converges. If aₙ ≥ bₙ ≥ 0 and ∑bₙ diverges, then ∑aₙ diverges. 10.6

Limit Comparison Test

If a n > 0, b n > 0 and $$\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=\ L$$ where L is finite and positive, then $$\sum_{n=1}^{\infty}a_n$$ and $$\sum_{n=1}^{\infty}b_n$$ either BOTH converge or diverge. 10.6

Alternating Series Test for Convergence

If terms alternate in sign, decrease in magnitude, and approach 0, the series converges. 10.7

Ratio Test for Convergence

For ∑a n , if $$\lim_{n\rightarrow\infty}\ \frac{a_{n+1}}{a_n}=L$$ : Converges if L < 1, Diverges if L > 1, Inconclusive if L = 1. 10.8

Determining Absolute or Conditional Convergence

Absolutely convergent if ∑|aₙ| converges. Conditionally convergent if ∑aₙ converges but ∑|aₙ| diverges. Divergent if both ∑|aₙ| and ∑aₙ diverge. 10.9

Nth Taylor Polynomial

$$P_n\left(x\right)=f\left(c\right)+f$$ $$\frac{f$$

Same as Taylor, but c = 0 $$P_n\left(x\right)=f\left(0\right)+f$$ $$...+\frac{f^n\left(0\right)x^n}{n!}$$

Coefficient of a Taylor Polynomial

$$\frac{f^{\left(n\right)}\left(c\right)}{n!}$$

Sequences and Series Flashcards

Student preview

23 questions

Show all answers

1.

FLASHCARD QUESTION

Front

nth Term of a Geometric Sequence

Back

Given by: a_n = a₁ r^n-1, where a₁ is the first term and r is the common ratio.
Or a_n= a₀rⁿ
Or a_n= a₂r^n-2

10.2

2.

FLASHCARD QUESTION

Front

Geometric Infinite Series Convergence

Back

3.

FLASHCARD QUESTION

Front

Geometric Infinite Series Divergence

Back

A geometric series diverges if |r| ≥ 1.

10.2

4.

FLASHCARD QUESTION

Front

Nth Term Test for Divergence

Back

5.

FLASHCARD QUESTION

Front

Integral Test for Convergence

Back

6.

FLASHCARD QUESTION

Front

P-Series Convergence

Back

7.

FLASHCARD QUESTION

Front

Harmonic Series

Back

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