To prove: The Maclaurin series for the given function converges for all real x using remainder estimation theorem
Explanation of Solution
Given:
The given function is
Proof:
As,
So, to show that the Maclaurin series for
Now,
Using Remainder theorem, it states that
Here M and r are constants,
Here, a = 0 as we are dealing with Maclaurin series
Since
Either
Which means that
Then,
Since,
Also
Since factorials increase faster than exponentials. This can also be done using the Ratio test.
So, conditions are satisfied
Hence proved that
Chapter 10 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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