How To Sum a Series
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Muhammad Yusuf
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The concept of an infinite sum is mysterious and intriguing.
How can you add up an infinite number of terms? Yet, in some contexts, we are
led to the contemplation of an infinite sum quite naturally. For example,
consider the calculation of a decimal expansion for 1/3. The long division algorithm
generates an endlessly repeating sequence of steps, each of which adds one more
3 to the decimal expansion. We imagine the answer therefore to be an endless
string of 3’s, which we write 0.333. . .. In essence we are defining the
decimal expansion of 1/3 as an infinite sum
1/3 = 0.3 + 0.03 + 0.003 + 0.0003 + ...
For another example, in a modification of Zeno’s paradox,
imagine partitioning a square of side 1 as follows: first draw a diagonal line
that cuts the square into two triangular halves, then cut one of the halves in
half, then cut one of these halves in half, and so on ad infinitum. (See Figure
1.) Then the area of the square is the sum of the areas of all the pieces,
leading to another infinite sum
1 = ½ + ¼ +
1/16 + 1/32 + ...
For continued text, you see in [here]
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