Die Hard Fan Of This Sequence!

Calculus Level 3

Let { a n } n ∈ N { \left\{ { a }_{ n } \right\} }_{ n \in \mathbb N } be a sequence of positive numbers converging to a a and assume that b b is a positive number such that b < a b < a .

Will the series converge or diverge?

∑ n = 1 ∞ n ! b n ( b + a 1 ) ( b + a 2 ) ⋅ ⋅ ⋅ ( b + a n ) \large\ \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { n!{ b }^{ n } }{ \left( b + { a }_{ 1 } \right) \left( b + { a }_{ 2 } \right) \cdot \cdot \cdot \left( b + { a }_{ n } \right) } }

It converges It diverges

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Priyanshu Mishra by Priyanshu Mishra , 20, India

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1 solution

Leonel Castillo
May 27, 2018

∑ n = 1 ∞ n ! b n ( b + a 1 ) . . . ( b + a n ) = ∑ n = 1 ∞ n ! ( 1 + a 1 b ) . . . ( 1 + a n b ) \sum_{n=1}^{\infty} \frac{n! b^n}{(b + a_1)...(b + a_n)} = \sum_{n=1}^{\infty} \frac{n!}{(1 + \frac{a_1}{b})...(1 + \frac{a_n}{b})} Let's define the sequence c n = a n b → a b > 1 c_n = \frac{a_n}{b} \to \frac{a}{b} > 1 to rewrite the series as ∑ n = 1 ∞ n ! ( 1 + c 1 ) . . . ( 1 + c n ) \sum_{n=1}^{\infty} \frac{n!}{(1 + c_1)...(1 + c_n)} . Applying the ratio test, ∣ n + 1 1 + c n + 1 ∣ → ∞ \left| \frac{n+1}{1 + c_{n+1}}\right| \to \infty because the numerator goes to infinity while the denominator converges to a finite positive number. Thus the series diverges.

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