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for finite term, the sum rule for differentiation is possible. my question is, what about infinite series? is there any special condition?

for finite term, the sum rule for differentiation is possible. my question is, what about infinite series?
is there any special condition?

Grade:

1 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
Hello student,
You are right that for finite series, we can simply use the sum rule for differentiation. But, in case of infinite series, one has to be a bit cautious about the general conditions under which an infinte series can be differentiated.
We know that the set of all formal power series whose coefficients are either form reals or complex numbers forms a ring under addition and term by term multiplication. This is simple algebra and hence is not affected by whether the sum converges or not.
In this case, hence, we can simply regard an infinite series as an infinite polynomial and so can add or differentiate terms where differentiation is simple derivation of terms. But, the situation is a bit different in cases where convergence of series matters. In that case, the central theorem plays a great role. It states:
Suppose we have a power series ∑{tnzn}. Then this series and its derivative both will have the same radius of convergence.
Now, therefore, the function s associated with ∑{tnzn} is differentiable in the disc of convergence, and the function represented by D(∑{tnzn}) agrees with f′ on the disc of convergence.
The main points that come to our notice from this is (you can also refer the proof which is easily available but is a bit lengthy):
When we take the differentiation of a power series that converges in some specific radii of convergence gives a series that also converges to the derivative of the function of original power sereis and both the series have the same radii of convergence.
Hence, it is almost always possible to differentiate a power series in the form of a ring although at times it may not yield a valid result in respect of calculus.

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