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## 'Harmonically' printed from http://nrich.maths.org/

Why do this problem?
With a hint about taking partitions of 1, 2, 4, 8 ... terms,
learners can discover that, unlikely as it may seem, although the
terms of the series get smaller and smaller, the sum of the series
grows to infinity.

The second part requires viewing the series as a sum of areas of
rectangles of unit width under the graph of $y={1\over x}$ and
doing this will reinforce the basic ideas of integration.

Possible approach
Make this a class effort and encourage discussion? Ask what they
think will happen to the series? Do they think the sum can grow
very big? By asking the key questions below the teacher can help
the class to investigate this important series.

The second part of the question could also be done by the class
working together and discussing the connection between the sum of
the series and the integral.

Key questions
What is the smallest term in each partition?

Can you find a lower bound for the sum of each partition?

How many partitions are needed to get a sum greater than 10?

What will the last term of the series be with this number of
partitions?

Now what about a sum greater than 100?

Possible extension
Overarch