Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 17/en

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Warm-up-exercises

Exercise

Compute the first five terms of the Cauchy product of the two convergent series


Exercise

Keep in mind that the partial sums of the Cauchy product of two series are not the product of the partial sums of the two series.


Exercise

Let and be two power series absolutely convergent in . Prove that the Cauchy product of these series is exactly


Exercise

Let  , . Determine (in dependence of ) the sum of the two series


Exercise

Let

be an absolutely convergent power series. Compute the coefficients of the powers in the third power


Exercise

Prove that the real function defined by the exponential

has no upper limit and that is the infimum (but not the minimum) of the image set.[1]


Exercise

Prove that for the exponential function

the following calculation rules hold (where and ).

  1. .
  2. .
  3. .
  4. .


Exercise

Prove that for the logarithm to base the following calculation rules hold.

  1. We have and , ie, the logarithm to base is the inverse to the exponential function to the base .
  2. We have
  3. We have for .
  4. We have


Exercise

A monetary community has an annual inflation of . After what period of time (in years and days), the prices have doubled?


Exercise

Let . Show that




Hand-in-exercises

Exercise (3 points)

Compute the coefficients of the power series , which is the Cauchy product of the geometric series with the exponential series.


Exercise (4 points)

Let

be an absolutely convergent power series. Determine the coefficients of the powers in the fourth power


Exercise (5 points)

For and let

be the remainder of the exponential series. Prove that for the remainder term estimate

holds.


Exercise (3 points)

Compute by hand the first digits in the decimal system of


Exercise (4 points)

Prove that the real exponential function defined by the exponential series has the property that for each the sequence

diverges to .[2]


Exercise (6 points)

Let

be a continuous function , with the property that
for all . Prove that is an exponential function, i.e. there exists a such that .




Fußnoten
  1. From the continuity it follows that is the image set of the real exponential function.
  2. Therefore we say that the exponential function grows faster than any polynomial function.



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