What is infty frac 1 infty

3.8. Series, power series and integrals

Every series can be interpreted as an improper integral:

\ (\ sum \ limits_ {n = 0} ^ \ infty a_n \) \ (= \ int \ limits_0 ^ {+ \ infty} T (x) \, \ diff x \)

for the through

\ (T (x): = a_n \) \ (\ quad \ text {if $ n \ le x \ lt n + 1 $} \)

defined "staircase function" \ (T \).

The function \ (T \) can be integrated on every restricted interval (according to 3.5.5.)

There is an interactive display for these step functions and the following comparison criterion.

3.8.1. Integral comparison criterion.

The function \ (f \ colon [m, + \ infty) \ to \ RR \) is positive and monotonically decreasing, with \ (m \ in \ NN \).

Then \ (\ sum \ limits_ {k = m} ^ \ infty f (k) \) and \ (\ int \ limits_m ^ {+ \ infty} f (x) \, \ diff x \) have the same convergence behavior.

Proof as a training unit in inequalities.

For all \ (k \ ge m \) and all \ (x \ in [k, k + 1] \) the following applies:

\ (f (k) \ ge f (x) \) \ (\ ge f (k + 1) \),

so

\ (f (k) \ ge \ int \ limits_k ^ {k + 1} f (x) \, \ diff x \) \ (\ ge f (k + 1) \).

This applies

\ (\ sum \ limits_ {k = m} ^ N f (k) \ ge \ int \ limits_m ^ {N + 1} f (x) \, \ diff x \) \ (\ ge \ sum \ limits_ {k = m + 1} ^ {N + 1} f (k) \)

and

\ (\ sum \ limits_ {k = m} ^ \ infty f (k) \ ge \ int \ limits_m ^ {+ \ infty} f (x) \, \ diff x \) \ (\ ge \ sum \ limits_ { k = m + 1} ^ {\ infty} f (k) \)

follows.

3.8.2. Example.

For \ (\ alpha \ gt 1 \) the series \ (\ sum \ limits_ {k = 1} ^ \ infty \ frac1 {k ^ \ alpha} \) converges, because the integral \ (\ int \ limits_ {1} ^ {+ \ infty} \ frac1 {x ^ \ alpha} \, \ diff x \) converges (see 3.7.8).

Caution: The limit values ​​\ (\ sum \ limits_ {k = 1} ^ \ infty \ frac1 {k ^ \ alpha} \) and \ (\ int \ limits_ {1} ^ {+ \ infty} \ frac1 {x ^ \ alpha} \, \ diff x \) are different!

E.g. for \ (\ alpha = 2 \) we know:

On the one hand it is \ (\ sum \ limits_ {k = 1} ^ \ infty \ frac1 {k ^ 2} = \ frac {\ pi ^ 2} 6 \) (see 1.8.3), but on the other hand it applies

\ (\ int \ limits_ {1} ^ {+ \ infty} \ frac1 {x ^ 2} \, \ diff x \) \ (= \ lim \ limits _ {\ beta \ to + \ infty} \ left [\ frac { -1} x \ right] _1 ^ \ beta \) \ (= \ lim \ limits _ {\ beta \ to + \ infty} \ frac {-1} \ beta - (-1) \) \ (= 1 \).

3.8.3. Comment.

The limit value criterion 3.7.11 is carried over to series with 3.8.1.

Particularly important series are power series.

3.8.4. Sentence.

Let \ (f (x) = \ sum \ limits_ {k = 0} ^ \ infty a_k \, (x-x_0 ^ {}) ^ k \) be a power series with a radius of convergence \ (\ rho \).

Then \ (f \) in the interval \ ((x_0 ^ {} - \ rho, x_0 ^ {} + \ rho) \) may be integrated or differentiated in terms of terms:

For all \ (x \ in (x_0 ^ {} - \ rho, x_0 ^ {} + \ rho) \) applies

\ (\ int \ limits_ {x_0 ^ {}} ^ xf (t) \, \ diff t \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {a_k} {k + 1} \ , (x-x_0 ^ {}) ^ {k + 1} \) \ (= \ sum \ limits_ {j = 1} ^ \ infty \ frac {a_ {j-1}} {j} \, (x- x_0 ^ {}) ^ {j} \)

or.

\ (f '(x) \) \ (= \ sum \ limits_ {k = \ alert {1}} ^ \ infty k \, a_k \, (x-x_0 ^ {}) ^ {k-1} \) \ (= \ sum \ limits _ {\ ell = 0} ^ \ infty (\ ell + 1) \, a _ {\ ell + 1} \, (x-x_0 ^ {}) ^ {\ ell} \).

The uniform convergence of the power series is used to prove this, we leave this to the mathematicians.

3.8.5. Example.

The function \ (f \ colon \ RR \ to \ RR \ colon x \ mapsto \ E ^ {\ alert {(} - x ^ 2 \ alert {)}} \) has the power series representation

\ (\ E ^ {\ alert {(} - x ^ 2 \ alert {)}} \) \ (= \ exp (\ color {blue} {- x ^ 2}) \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ dfrac {(\ color {blue} {- x ^ 2}) ^ k} {k!} \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {(-1) ^ k} {k!} \, X ^ {2 \, k} \),

so the coefficients are

\ (a_n = \) \ (\ begin {cases} \ frac {(- 1) ^ {\ frac n2}} {\ frac n2!} & \ text {if $ n $ even,} \ \ qquad 0 & \ text {otherwise.} \ end {cases} \)

Therefore, according to Theorem 3.8.4,

\ (\ int \ limits_0 ^ xf (t) \, \ diff t \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {(- 1) ^ k} {k! \, (2k +1)} \, x ^ {2 \, k + 1} \).

3.8.6. Remarks.

The function \ (\ E ^ {(- x ^ 2)} \) is not elementary integrable:

The antiderivative (which we described as a power series in 3.8.5) cannot be represented by an algebraic combination of the "elementary functions" (polynomials, \ (\ exp \), \ (\ ln \), trigonometric functions).

The power series in 3.8.5 are the Taylor series of the corresponding functions in the expansion point \ (x_0 ^ {} = 0 \).

3.8.7. Example.

Another function that cannot be integrated in elementary terms is

\ (f \ colon \ RR \ setminus \ {0 \} \ to \ RR \ colon \) \ (x \ mapsto \ begin {cases} \ frac {\ sin x} x & \ text {if $ x \ ne0 $ } \ 0 & \ text {if $ x = 0 $.} \ End {cases} \)

(This function is continuous at \ (x_0 = 0 \) - you can see that with 1.12.5, or with the rule of l'Hospital, but easiest with the power series representation.)

The antiderivative has its own name:

The integral sine \ (\ operatorname {Si} \) is given by

\ (\ operatorname {Si} x \) \ (: = \ int \ limits_ {0} ^ x \ frac {\ sin t} t \, \ diff t \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {(- 1) ^ k} {(2 \, k + 1)! \, (2 \, k + 1)} \, x ^ {2 \, k + 1} \).

In the following sketch the function \ (\ operatorname {Si} \) was actually approximated by partial sums of this power series (i.e. Taylor polynomials of \ (\ operatorname {Si} \)) & emdash; you can also see that this only works well in sufficiently small intervals ...

To see that \ (\ operatorname {Si} \) is an antiderivative of \ (\ frac {\ sin x} x \), one describes \ (\ sin x \) by a power series:

\ (\ sin x \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {(- 1) ^ k} {(2 \, k + 1)!} \, x ^ {2 \ , k + 1} \);

so

\ (\ dfrac {\ sin x} x \) \ (= \ sum \ limits_ {k = 0} ^ \ infty \ frac {(- 1) ^ k} {(2 \, k + 1)!} \, x ^ {2 \, k} \).

One makes sure [cf. 1.14.20] that the radius of convergence of these series is \ (+ \ infty \).

Now link-wise integration provides the assertion.

3.8.8. Example.

The geometric series describes \ (\ dfrac1 {1-x} \) \ (= \ sum \ limits_ {k = 0} ^ \ infty x ^ k \) for \ (| x | \ lt 1 \):

Term-wise differentiation of the right-hand side and simple derivation of the left-hand side yield (for \ (| x | \ lt 1 \)):

\ (\ ds \ frac1 {(1-x) ^ 2} \) \ (= \ sum \ limits_ {k = \ alert {1}} ^ \ infty k \, x ^ {k-1} \) \ ( = \ sum \ limits_ {j = 0} ^ \ infty (j + 1) \, x ^ {j} \).

So this approach is (sometimes) suitable for determining the limit of a series.