Singular Inner Functions

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The product n rk converges if and only if there exists r 0 such that n lim n n rk r. k1 By continuity of log on (0, 1], this occurs if and only if

or, equivalently,

lim n

n log n rk log r, k1 lim n n log rk log r. k1 This is the same as convergence of }, log rk . Since

lim x1 log x 1 x

1, the limit comparison test shows that the above series converges if and only if

'(1 rk ) k1 converges. When a series converges, its tail goes to 0. Similarly, when an innite product converges, its tail goes to 1. Theorem . Let 0 rk 1 for all k. If n

rk converges, then ( n n rk km1 converges to 1 as n and m approach infinity.

Proof. Observe that the above sequence is just k1 rk k1 rk and hence,...

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