A **series** is just the sum of some set of terms of a sequence. For example, the sequence 2, 4, 6, 8, ... has partial sums of 2, 6, 12, 20, ... These partial sums are each a **finite series**. The *n*th partial sum of a sequence is usually called *S _{n}*. If the sequence being summed is

See About the calculus applets for operating instructions. |

*s*_{1} and *d* set to generate odd integers. On this applet, the sequence is shown as rectangles of width 1, somewhat reminiscent of a
Riemann sum. The first term of this sequence is 1 so the first rectangle is 1 by 1 wide. The second term in our example is 3 so the next rectangle is 3 high by 1 wide, the third term is 5 high by 1 wide and so on.

The dots on this graph represent the different finite series, each being the sum of the terms of the sequence up to that point. From a visualization standpoint, think of the height of each dot as being the total area of all the rectangles to the left of the dot. Hence the first dot is at (1,1), the second dot is at (2,3), the third is at (3,5), etc. Input boxes and sliders are provided to allow you to change *s*_{1} and *d*. The table on the left gives some large values of the sequence and the series, clearly showing that the infinite series diverges.

Select the second example from the drop down menu, showing the geometric sequence *s _{n}* =

Select the third example, showing the sequence* s _{n}* = (

Select the fourth example, showing the **harmonic series** defined by
The sequence converges to zero, but looking at the table, it isn't clear whether the series converges or not. Also, the points of the series in the graph resemble the graph of *y* = ln(*x*), which we know doesn't converge. In fact, the harmonic series is divergent; it keeps growing without bound, albeit slowly.

Note that the nth term divergence test says only that if the sequence converges to a non-zero number, then the series diverges. It does not say what happens if the sequence does converge to zero. In that case the series may converge or diverge, depending on how fast the sequence converges to zero. The harmonic sequence does converge to zero, but it just doesn't do it fast enough for the harmonic series to also converge.

Select the fifth example, showing the **p-series** defined by
With *p* = 1 we get the harmonic series from the previous example, which we know does not converge. Move the *p* slider to see what happens for *p* > 1, 0 < *p* < 1, and *p* < 0. For which of these cases does it appear that the p-series converges? You should notice that a *p-*series converges for *p* > 1 and diverges otherwise.

- Sequences
- Series
- Integral Test
- Comparison Test
- Limit Comparison Test
- Ratio Test
- Alternating Series and Absolute Convergence
- Power Series & Interval of Convergence
- Taylor Series & Polynomials
- Lagrange Remainder

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