Infinite Series and the Geometric Series
Define the sum of a series as the limit of its partial sums, work out the geometric and telescoping cases exactly, and use the term test to rule out convergence.
What a learner can do afterwards
- Write the partial sum of a telescoping series and take its limit
- State the condition under which a geometric series has a finite sum
- Show that terms not tending to zero forces divergence, and that the converse fails
1 · Read
A series is the limit of its partial sums: add the first N terms, then watch the totals as N grows. Some series collapse: 1 over n times n plus 1 splits into 1 over n minus 1 over n plus 1, so neighbours cancel. The Nth partial sum is N over N plus 1, which tends to 1, and that limit is the sum.
A geometric series multiplies by a fixed ratio each step, and it settles exactly when the ratio sits inside the unit circle. With first term a and ratio r, the sum is a over 1 minus r. So 1 plus 1 over 2 plus 1 over 4 and on gives 1 over 1 minus 1 over 2, which is 2. And 4 plus 4 over 3 plus 4 over 9 and on gives 4 over 1 minus 1 over 3, which is 6.
Rafi eyes a series whose nth term tends to 5 and declares divergence at once, and he is right: terms that refuse to vanish break convergence. But passing this filter proves nothing, since the harmonic series has terms tending to zero yet its totals grow without bound. Zero-limit terms are required for convergence but never enough on their own.
Run three checks in order on any new series. Write partial sums first when neighbours might cancel. Check for a constant ratio and use first term over 1 minus ratio. Apply the term test as a quick divergence filter, and remember that silence from it settles nothing.
Partial sums define the sum, geometric series need a small ratio, and vanishing terms never promise convergence.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.