8.Suppose that the seriesPcn has radius of convergence 5 and the seriesPdn has radius of convergence 8. What is the radius of convergence of the seriesP(cn dn)xn

Answer and Step-by-step explanation:

Solution:

Suppose the series:

                                     Ʃcnxn

has the radius of convergence 5,

and the series:

                        Ʃdnxn

has the radius of convergence 8.

find the radius of convergence of series:

Ʃ ( cn + dn) xn = Ʃcnxn +  Ʃdnxn

The radius of convergence of the cn series falls within the radius of dn, series.

So, both will converges for at least |x| < 5.

If we say, x = 5.5, then we get the sum of divergent series plus a convergent series, for which overall result is divergent.

For example:

∞ + a constant, gives result in ∞.

The radius of convergence is the minimum radius for which both series converges,

Which is 5.

Hence, R = 5.

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