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- interest compounded periodically
------- interest compounded continuously
When interest is compounded continously, this means letting the number of periods in a year increase without bound.
In other words, we find the limit, as n tends to infinity of
Binomial theorem
Applying the Binomial theorem to
, we get
as
,
, giving
and the above is the Maclaurin series for
i.e.
as
We wanted to find the limit, as n tends to infinity of
and we have the limit, as n tends to infinity, of
which is
. We substitute
for
.
So for continuous compounding we can write,
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