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Convergence of series

Divergence, ratio, comparison and integral tests — and the harmonic series.

An infinite series converges when its partial sums approach a limit. The terms going to zero is necessary but not enough — the harmonic series 1 + ½ + ⅓ + … diverges even though its terms vanish. The ratio test settles most series with factorials or powers; comparison and the integral test handle the p-series family.

Shembulli i punuar: does 1/n^2 for n = 1 to oo converge

Calculus · Series-And-Sums

Convergence of n^(-2)

\frac{1}{n^{2}}

Hap pas hapi

  1. \sum_{n=1}^{\infty} \frac{1}{n^{2}}

    Does this series converge?

  2. \lim_{n\to\infty} \frac{1}{n^{2}} = 0

    Divergence test: if the terms do not go to 0 the series cannot converge.

  3. \lim \left|\frac{a_{k+1}}{a_k}\right| = \lim \left|{\frac{n^{2}}{\left(n + 1\right)^{2}}}\right| = 1

    Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.

  4. = \frac{\pi^{2}}{6}

    It converges, and the CAS knows the sum.

Zbulo përgjigjen
\text{converges to } \frac{\pi^{2}}{6}

Symbols used here

\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varepsilon,\ \delta
epsilon, delta
Small positive tolerances in the definition of a limit.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.

How to: Convergence of series

  1. Does this series converge?
  2. Divergence test: if the terms do not go to 0 the series cannot converge.
  3. Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.
  4. It converges, and the CAS knows the sum.

Questions people ask

What is the ε–δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on — infinitely many halves.

Provo timen.

Më shumë në Real Analysis

Sequences and their limitsImproper integralsTaylor approximation and errorFunctions of a complex variableContinuity and differentiability, rigorouslySequences of functions and uniform convergence