Sequence & Series Formulas — Complete Reference A complete reference of sequence and series formulas. Covers arithmetic and geometric sequences, partial and infinite sums, the most common power series, and convergence tests used in calculus.
Arithmetic Sequences nth Term
a n = a 1 + ( n − 1 ) d a_n = a_1 + (n - 1)\,d a n = a 1 + ( n − 1 ) d Recursive Form
a n = a n − 1 + d a_n = a_{n-1} + d a n = a n − 1 + d Common Difference
d = a n + 1 − a n d = a_{n+1} - a_n d = a n + 1 − a n Sum of First n Terms
S n = n 2 ( a 1 + a n ) S_n = \frac{n}{2}(a_1 + a_n) S n = 2 n ( a 1 + a n ) Sum (using d)
S n = n 2 [ 2 a 1 + ( n − 1 ) d ] S_n = \frac{n}{2}\left[2 a_1 + (n - 1)\,d\right] S n = 2 n [ 2 a 1 + ( n − 1 ) d ] Geometric Sequences & Series nth Term
a n = a 1 ⋅ r n − 1 a_n = a_1 \cdot r^{n-1} a n = a 1 ⋅ r n − 1 Common Ratio
r = a n + 1 a n r = \frac{a_{n+1}}{a_n} r = a n a n + 1 Sum of First n Terms
S n = a 1 ⋅ 1 − r n 1 − r ( r ≠ 1 ) S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad(r \ne 1) S n = a 1 ⋅ 1 − r 1 − r n ( r = 1 ) Infinite Geometric Sum
S = a 1 1 − r ( ∣ r ∣ < 1 ) S = \frac{a_1}{1 - r} \quad(|r| < 1) S = 1 − r a 1 ( ∣ r ∣ < 1 ) Repeating Decimal
0. a b ‾ = ab 99 , etc. 0.\overline{ab} = \frac{\text{ab}}{99},\ \text{etc.} 0. ab = 99 ab , etc. Special Sums Sum of First n Integers
∑ k = 1 n k = n ( n + 1 ) 2 \sum_{k=1}^{n} k = \frac{n(n+1)}{2} k = 1 ∑ n k = 2 n ( n + 1 ) Sum of First n Squares
∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} k = 1 ∑ n k 2 = 6 n ( n + 1 ) ( 2 n + 1 ) Sum of First n Cubes
∑ k = 1 n k 3 = [ n ( n + 1 ) 2 ] 2 \sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2 k = 1 ∑ n k 3 = [ 2 n ( n + 1 ) ] 2 Sum of Constant
∑ k = 1 n c = c n \sum_{k=1}^{n} c = c n k = 1 ∑ n c = c n Power & Maclaurin Series Geometric Series
1 1 − x = ∑ n = 0 ∞ x n ( ∣ x ∣ < 1 ) \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \quad(|x| < 1) 1 − x 1 = n = 0 ∑ ∞ x n ( ∣ x ∣ < 1 ) Exponential
e x = ∑ n = 0 ∞ x n n ! e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} e x = n = 0 ∑ ∞ n ! x n Sine
sin x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} sin x = n = 0 ∑ ∞ ( 2 n + 1 )! ( − 1 ) n x 2 n + 1 Cosine
cos x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} cos x = n = 0 ∑ ∞ ( 2 n )! ( − 1 ) n x 2 n ln(1 + x)
ln ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 x n n ( ∣ x ∣ < 1 ) \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \quad(|x| < 1) ln ( 1 + x ) = n = 1 ∑ ∞ n ( − 1 ) n + 1 x n ( ∣ x ∣ < 1 ) Arctangent
tan − 1 x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 ( ∣ x ∣ ≤ 1 ) \tan^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \quad(|x| \le 1) tan − 1 x = n = 0 ∑ ∞ 2 n + 1 ( − 1 ) n x 2 n + 1 ( ∣ x ∣ ≤ 1 ) Binomial Series
( 1 + x ) k = ∑ n = 0 ∞ ( k n ) x n ( ∣ x ∣ < 1 ) (1 + x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n \quad(|x| < 1) ( 1 + x ) k = n = 0 ∑ ∞ ( n k ) x n ( ∣ x ∣ < 1 ) Taylor Series & Remainder Taylor Series at x = a
f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n f ( x ) = n = 0 ∑ ∞ n ! f ( n ) ( a ) ( x − a ) n f ( x ) = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! x n f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n f ( x ) = n = 0 ∑ ∞ n ! f ( n ) ( 0 ) x n Lagrange Remainder
R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1} R n ( x ) = ( n + 1 )! f ( n + 1 ) ( c ) ( x − a ) n + 1 ∣ R n ∣ ≤ ∣ a n + 1 ∣ |R_n| \le |a_{n+1}| ∣ R n ∣ ≤ ∣ a n + 1 ∣ Convergence Tests nth Term Test (Divergence)
lim n → ∞ a n ≠ 0 ⟹ ∑ a n diverges \lim_{n \to \infty} a_n \ne 0 \implies \sum a_n \text{ diverges} n → ∞ lim a n = 0 ⟹ ∑ a n diverges Ratio Test
L = lim n → ∞ ∣ a n + 1 a n ∣ ; L < 1 ⇒ converges L = \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|;\ L < 1 \Rightarrow \text{converges} L = n → ∞ lim a n a n + 1 ; L < 1 ⇒ converges Root Test
L = lim n → ∞ ∣ a n ∣ n ; L < 1 ⇒ converges L = \lim_{n \to \infty} \sqrt[n]{|a_n|};\ L < 1 \Rightarrow \text{converges} L = n → ∞ lim n ∣ a n ∣ ; L < 1 ⇒ converges Integral Test
∑ n = 1 ∞ a n and ∫ 1 ∞ f ( x ) d x converge together \sum_{n=1}^{\infty} a_n \text{ and } \int_1^\infty f(x)\,dx \text{ converge together} n = 1 ∑ ∞ a n and ∫ 1 ∞ f ( x ) d x converge together p-Series
∑ 1 n p converges ⟺ p > 1 \sum \frac{1}{n^p} \text{ converges} \iff p > 1 ∑ n p 1 converges ⟺ p > 1 Alternating Series Test
∑ ( − 1 ) n a n converges if a n ↓ 0 \sum (-1)^n a_n \text{ converges if } a_n \downarrow 0 ∑ ( − 1 ) n a n converges if a n ↓ 0 