Binomial Sequence

If b > 1 then the sequence converges to zero for any positive integer k. In fact, this is still true if k is replaced by any real number.

Proof:

Note that both numerator and denominator tend to infinity. Our goal will be to show that the denominator grows faster than the k-th power of n, thereby 'winning' the race to infinity and forcing the whole expression to tend to zero.

The name of this sequence indicates that we might try to use the binomial theorem for this. Indeed, define x such that

b = 1 + x

Since b > 1 we know that x > 0. Therefore, each term in the binomial theorem is positive, and we can use the (k+1)-st term of that theorem to estimate:

for any k+1 n

Let n = 2k + 1, or equivalently, k = (n-1)/2. Then n - k = n - (n-1)/2 = (n+1)/2 > n/2, so that each of the expressions n, n-1, n-2, ..., n - k is greater than n/2. Hence, we have that

But then, taking reciprocals, we have:

But this expression is true for all n > 2k + 1 as well, so that, with k fixed, we can take the limit as n approaches infinity and the right hand side will approach zero. Since the left-hand side is always greater than or equal to zero, the limit of the binomial sequence must also be zero.

If k is replaced by any real number, the proof is similar, and is left as an exercise.

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