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Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?


Age 16 to 18
Challenge Level

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