Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 17/en
- 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 ).
- .
- .
- .
- .
Exercise
Prove that for the logarithm to base the following calculation rules hold.
- We have and , ie, the logarithm to base is the inverse to the exponential function to the base .
- We have
- We have for .
- 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
- Fußnoten
- ↑ From the continuity it follows that is the image set of the real exponential function.
- ↑ Therefore we say that the exponential function grows faster than any polynomial function.
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