1.6 Infinite series | Sequences and series

1.6 Infinite series (EMCF3) So far we have been working only with finite sums, meaning that whenever we determined the sum of a series, we only considered the sum of the first \(n\) terms. We now consider what happens when we add an infinite number of terms together. Surely if we sum infinitely many numbers, no matter how small they are, the answer goes to infinity? In some cases the answer does indeed go to infinity (like when we sum all the positive integers), but surprisingly there are some cases where the answer is a finite real number. Sum of an infinite series Cut a piece of string \(\text{1}\) \(\text{m}\) in length. Now cut the piece of string in half and place one half on the desk. Cut the other half in half again and put one of the pieces on the desk. Repeat this process until the piece of string is too short to cut easily. Draw a diagram to illustrate the sequence of lengths of the pieces of string. Can this sequence be expressed mathematically? Hint: express the shorter len...