p≤1 diverge

p>1 convergence

q≥1 diverge

q<1 convergence

Orthogonality of systems of trigonometric functions

Functions with known power series expansions (must remember)

Complex quantity->Simple quantity superposition

Application scope

condition

Sort by ascending power

Each term is a function of a power of x

definition

how to research

Abel's theorem

Possible cases of convergence

How to get it

Notice

If the coefficients of two power series are the same, then their convergence regions are the same (convergence radius, convergence interval)

premise

nature

Power series and functions have good properties within their convergence region

Is it possible to expand a function into a power series?

Can it be expanded (is there a power series that can be equated with function f(x))

If it can be expanded, what is the formula for power series expansion (study first)

If f(x) can be expanded into a power series, then the power series can only be the Taylor series of f(x)

f(x)nth order Lagrangian remainder Rn(x)->0(n->∞)<->f(x) can be expanded into a Taylor series

direct expansion method

indirect expansion method

Taylor formula

Taylor series

Remainder->0(n->∞), the sum of the first n terms of the infinite series converges to f(x) Explain that f(x) is the sum function corresponding to Taylor series (f(x) can be expanded into Taylor series)

Expand the power series you are looking for (write down a few to see), and compare the power series expansions of the existing sum functions

Compare with the existing formula to make up the existing formula (match the number)

Perform Fourier series expansion on the function f(x) with period 2l defined on any interval

For the convenience of research, generally only the interval [-l, l] that is symmetrical about the origin and has a length of 2l is taken for research, and the corresponding conclusions can be obtained based on the periodic continuation of the remaining intervals.

1. Calculate the coefficients and write the Fourier series F(x)

2. Determine the domain of f(x)=F(x) according to the convergence theorem

f(x) is continuous or has a finite number of discontinuities of the first kind

f(x) has at most a limited number of extreme points (does not oscillate infinitely)

continuous points

discontinuity

Interval (one period) endpoint

step

How to do the questions

Partial sums of infinite series (finite)

Limits of Parts and Sequences (Limits)

Existence (core)

If present, value

Study infinite series using limits of parts and sequences

Not ±existence = no; not±not = not necessarily (limited rational arithmetic rule)

Convergent series add parentheses and still converge and the sum remains unchanged (sequence limit and partial sequence limit)

(Necessary condition for series convergence) Series convergence-->series general formula un=0(n->∞)

Remove, add or change finite terms of the series -> does not affect the convergence of the series but changes the sum of the series

Pay attention to the negative proposition

Absolute value series converges--must->original series converges

The original series converges but the absolute value series does not.

in conclusion

Negative term series (proposing that the sign becomes a positive term)

positive series

staggered series

Arbitrary term series