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This series is known as the Harmonic Series and it diverges but very, very slowly.

For example, the first 100 terms sum to 5.187...., the first 1000 terms to 7.486...., and the first 1000000 terms to 14.392....

There are many proofs of the divergence of this series and an internet search of Harmonic Series will no doubt find many of them.

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