Taylor series formulas

Taylor series. 14 formulas for GATE DA Calculus and Optimization, each with when to use it. Each is explained in the topic notes.

All 14 Taylor series formulas

TAYLOR SERIES ABOUT X = A

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n} = f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \dots

Valid for those $x$ where the series converges to $f (x)$ ; convergence is not automatic, so state the range of $x$ .

MACLAURIN SERIES (A = 0)

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( 0 )}{n !} x^{n}

The coefficient of $x^{n}$ is $f^{(n)} (0) / n!$ , so $f^{(n)} (0) = n! \times (coefficient of x^{n})$ .

EXPONENTIAL

e^{x} = n = 0 \sum \infty \frac{x ^{n}}{n !} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots

Converges for all real $x$ .

SINE

sin x = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!} = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots

Converges for all real $x$ ; only odd powers appear.

COSINE

cos x = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!} = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots

Converges for all real $x$ ; only even powers appear.

NATURAL LOGARITHM

ln (1 + x) = n = 1 \sum \infty \frac{( - 1 ) ^{n + 1} x ^{n}}{n} = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

Valid for $- 1 < x \leq 1$ . At $x = 1$ it gives $ln 2$ ; at $x = - 1$ it diverges.

BINOMIAL SERIES

(1 + x)^{m} = k = 0 \sum \infty (k m) x^{k} = 1 + m x + \frac{m ( m - 1 )}{2 !} x^{2} + \dots, (k m) = \frac{m ( m - 1 ) \dots ( m - k + 1 )}{k !}

For any real $m$ it is valid for $∣ x ∣ < 1$ and diverges for $∣ x ∣ > 1$ (radius 1) unless $m$ is a non-negative integer, in which case the series stops and is exact for all $x$ . At $x = \pm 1$ convergence depends on $m$ (for example, for $m > 0$ it also converges at both endpoints).

GEOMETRIC SERIES

\frac{1}{1 - x} = n = 0 \sum \infty x^{n} = 1 + x + x^{2} + \dots

Valid for $∣ x ∣ < 1$ . Replace $x$ by $- x$ for $1/ (1 + x) = 1 - x + x^{2} - \dots$ .

INVERSE TANGENT

tan^{- 1} x = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n + 1}}{2 n + 1} = x - \frac{x ^{3}}{3} + \frac{x ^{5}}{5} - \dots

Valid for $- 1 \leq x \leq 1$ ; obtained by integrating $1/ (1 + t^{2}) = 1 - t^{2} + t^{4} - \dots$ from $0$ to $x$ . At $x = 1$ it gives $π /4$ .

TANGENT (FIRST TERMS)

tan x = x + \frac{x ^{3}}{3} + \frac{2 x ^{5}}{15} + \dots

Valid for $∣ x ∣ < π /2$ ; only odd powers appear. The first two terms are usually enough for limits as $x \to 0$ .

RADIUS OF CONVERGENCE (RATIO TEST)

R = n \to \infty lim \frac{c _{n}}{c _{n + 1}} for n = 0 \sum \infty c_{n} (x - a)^{n}

When this limit exists (it may be $\infty$ ), the series converges absolutely for $∣ x - a ∣ < R$ and diverges for $∣ x - a ∣ > R$ ; the endpoints $∣ x - a ∣ = R$ must be checked separately. For a series with only odd or only even powers, apply the ratio test to consecutive non-zero terms instead.

TAYLOR'S THEOREM WITH LAGRANGE REMAINDER

f (x) = P_{n} (x) + R_{n} (x), R_{n} (x) = \frac{f ^{(n + 1)} ( c )}{( n + 1 )!} (x - a)^{n + 1} for some c strictly between a and x

Needs $f^{(n)}$ continuous on the closed interval between $a$ and $x$ and $f^{(n + 1)}$ to exist on the open interval (having $n + 1$ continuous derivatives is enough). Hence $∣ R_{n} (x) ∣ \leq \frac{M}{( n + 1 )!} ∣ x - a ∣^{n + 1}$ , where $M$ is the largest value of $∣ f^{(n + 1)} ∣$ on that interval.

ALTERNATING SERIES ERROR BOUND

∣ error after n terms ∣ \leq ∣ first omitted term ∣

Applies when, from the first omitted term on, the terms alternate in sign and decrease in magnitude to 0; the error then has the same sign as the first omitted term. Examples: $ln (1 + x)$ for $0 < x \leq 1$ ; $sin x$ , $cos x$ and $tan^{- 1} x$ for $∣ x ∣ \leq 1$ .

HIGHER-ORDER DERIVATIVE TEST

f^{'} (a) = f^{''} (a) = \dots = f^{(n - 1)} (a) = 0, f^{(n)} (a) \neq = 0 (n \geq 2)

Assume $f^{(n)}$ is continuous near $a$ . If $n$ is even: local minimum when $f^{(n)} (a) > 0$ , local maximum when $f^{(n)} (a) < 0$ . If $n$ is odd: no local extremum at $a$ . For $n = 2$ this is the second-derivative test.

Where this was asked: 2025 (1), 2026 (1). See the 2 previous-year questions on Taylor series. This shows what was asked, not what will be asked.

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Taylor series: the notes and 2 solved previous-year questions