Sequences & Series formulas

All 20 formulas in this topic

Arithmetic Sequencean=a1+(n−1)da_n = a_1 + (n - 1)dFind the nth term in a sequence with constant differences between consecutive terms, or determine how many steps it takes to reach a specific value.Arithmetic SeriesSn=n2(a1+an)S_n = \frac{n}{2}\big(a_1 + a_n\big)Find the sum of an arithmetic sequence without adding every term—just use the first term, last term, and how many terms there are.Geometric Sequencean=a1r n−1a_n = a_1 r^{\,n-1}The formula a_n = a_1 r^{n-1} finds any term in a geometric sequence, where each term is a constant multiple of the previous one.Geometric SeriesSn=a11−rn1−r,r≠1S_n = a_1 \frac{1 - r^{n}}{1 - r}, \quad r \neq 1Sums the first n terms of a geometric sequence—those where each term is the previous one times a constant—using a single formula instead of adding.Infinite Geometric SeriesS=a11−r,∣r∣<1S = \frac{a_1}{1 - r}, \quad |r| < 1Infinite Geometric Series gives you the sum of all terms in a geometric sequence when the common ratio lies strictly between negative one and one.Sum of First n Integers∑k=1nk=n(n+1)2\sum_{k=1}^{n} k = \frac{n(n+1)}{2}Add all integers from 1 to n using the formula \frac{n(n+1)}{2} instead of manually summing—much faster for large numbers.Sum of Squares∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}Sum of Squares gives you a formula to calculate the sum of the first n perfect squares instantly, without adding each term one by one.Sum of Cubes Formula∑k=1nk3=[n(n+1)2]2\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2The Sum of Cubes Formula calculates the sum of the first n cubes and shows that it equals the square of the sum of the first n integers.Recursive Formulaan=f(an−1),a1 givena_n = f(a_{n-1}), \quad a_1 \text{ given}A recursive formula defines each term using the previous one, helping you find patterns and generate sequences step-by-step.Fibonacci SequenceFn=Fn−1+Fn−2,F1=F2=1F_n = F_{n-1} + F_{n-2}, \quad F_1 = F_2 = 1The Fibonacci sequence is a series where each term equals the sum of the previous two terms; it models growth patterns found in nature.Sigma Notation∑k=mnak=am+am+1+⋯+an\sum_{k=m}^{n} a_k = a_m + a_{m+1} + \cdots + a_nSigma notation compresses long sums into a compact formula, letting you write and calculate the sum of many terms efficiently and accurately.Factorialn!=n(n−1)(n−2)⋯1,0!=1n! = n(n-1)(n-2)\cdots 1, \quad 0! = 1Factorial multiplies all positive integers up to n to find permutations. Use it to count the number of ways to arrange distinct objects in order.Telescoping Series∑k=1n(bk−bk+1)=b1−bn+1\sum_{k=1}^{n} \big(b_k - b_{k+1}\big) = b_1 - b_{n+1}A telescoping series simplifies sums where consecutive terms partially cancel out, leaving only the first and last terms to compute.Harmonic Series∑n=1∞1n=∞\sum_{n=1}^{\infty} \frac{1}{n} = \inftyThe Harmonic Series, sum of reciprocals of positive integers, diverges infinitely—a foundational benchmark for determining whether other series converge.p-Series Test∑n=1∞1np converges  ⟺  p>1\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges} \iff p > 1The p-Series Test determines convergence of series 1/n^p: the series converges if and only if the exponent p is strictly greater than 1.Ratio TestL=lim⁡n→∞∣an+1an∣,L<1  ⟹  convergesL = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|, \quad L < 1 \implies \text{converges}Determine whether an infinite series converges by checking if the ratio of consecutive terms approaches a value less than one.Integral Test∑n=1∞f(n) and ∫1∞f(x) dx share convergence\sum_{n=1}^{\infty} f(n) \text{ and } \int_1^{\infty} f(x)\,dx \text{ share convergence}Test series convergence by checking improper integral ∫₁^∞ f(x)dx when f is positive, continuous, and decreasing; the series ∑f(n) shares convergence behavior.Alternating Series Test∑(−1)nbn converges if bn↓0\sum (-1)^n b_n \text{ converges if } b_n \downarrow 0The Alternating Series Test determines whether a series with alternating signs converges when its terms decrease monotonically to zero.Taylor Seriesf(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^nA Taylor Series expresses a function as an infinite polynomial centered at a point, approximating smooth functions using just derivatives.Maclaurin Seriesf(x)=∑n=0∞f(n)(0)n!xnf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^nA Maclaurin Series expresses a function as an infinite polynomial using its derivatives at zero, letting you approximate functions without a calculator.

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