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19 results for “series”

Formulae (18)

Arithmetic Series Sum

S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence.

AlgebraIntermediate

Geometric Series Sum

S_n = a_1 \frac{1 - r^n}{1 - r}

Sum of the first n terms of a geometric sequence (r ≠ 1).

AlgebraIntermediate

Infinite Geometric Series

S = \frac{a_1}{1 - r}, \quad |r| < 1

Sum of an infinite geometric series that converges.

AlgebraAdvanced

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

Closed form for the sum of the first n cubes.

AlgebraAdvanced

Infinite Geometric Series

S = \frac{a}{1-r}, \; |r|<1

Sum of an infinite geometric series that converges.

AlgebraIntermediate

Finite Geometric Series

S_n = a\frac{1-r^n}{1-r}

Sum of the first n terms of a geometric series.

CalculusAdvanced

Taylor Series

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

Expands a function as an infinite polynomial around a.

CalculusAdvanced

Maclaurin Series for e^x

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}

Taylor series of the exponential at 0.

Electronics & CircuitsBasic

Resistors in Series

R_{eq} = R_1 + R_2 + \cdots

Series resistances add directly.

Electronics & CircuitsIntermediate

Capacitors in Series

\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots

Reciprocals of series capacitances add.

Electronics & CircuitsIntermediate

Voltage Divider

V_{out} = V_{in}\frac{R_2}{R_1 + R_2}

Output voltage from two series resistors.

Electronics & CircuitsBasic

LED Series Resistor

R = \frac{V_s - V_f}{I}

Resistor to safely drive an LED.

Electronics & CircuitsIntermediate

LED Series Resistor

R = \frac{V_s - V_f}{I}

Resistor to safely drive an LED.

EngineeringBasic

Resistors in Series

R_{eq} = R_1 + R_2 + \cdots

Total resistance of resistors in series.

Finance & InterestAdvanced

Present Value of Annuity

PV = P\frac{1 - (1+r)^{-n}}{r}

Present value of a series of equal payments.

Number TheoryBasic

Sum of First n Integers

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Closed form for the sum of the first n natural numbers.

Number TheoryIntermediate

Sum of First n Squares

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Closed form for the sum of the first n squares.

Number TheoryIntermediate

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

The sum of the first n cubes equals the square of the sum.

Math Tricks (1)

Sum of 1 to N Instantly

Gauss's formula: the sum of the first n whole numbers is n(n+1)/2 — no adding required.

View trick