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Asked Feb 26, 2020
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17. Prove the corollary to Theorem 5.24.
If A e Mpxn(C), define e^
lim Bm, where
Definition.
A?
Bm = I+ A+
2!
m!
(see Exercise 20). Thus e^ is the sum of the infinite series
А? АЗ
-+ ...
3!
I+A+:
2!
and Bm is the mth partial sum of this series. (Note the analogy with the power
series
a? a
e* - 1+ a +
2!
3!
which is valid for all complex numbers a.)
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17. Prove the corollary to Theorem 5.24. If A e Mpxn(C), define e^ lim Bm, where Definition. A? Bm = I+ A+ 2! m! (see Exercise 20). Thus e^ is the sum of the infinite series А? АЗ -+ ... 3! I+A+: 2! and Bm is the mth partial sum of this series. (Note the analogy with the power series a? a e* - 1+ a + 2! 3! which is valid for all complex numbers a.)

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