Note: The dots " We also show a proof using Algebra below. We often use Sigma Notation for infinite series.
Our example from above looks like:. Let's add the terms one at a time.
When the "sum so far" approaches a finite value, the series is said to be " convergent ":. When the difference between each term and the next is a constant, it is called an arithmetic series.
When the ratio between each term and the next is a constant, it is called a geometric series. This illustration may convince you that the tems converge on 1 3 :. Maybe you can try to prove it yourself? Try pairing up each plus and minus pair, then look up above for a series that matches.
Using integral calculus trust me that area is ln 2 :. The order of the terms can be very important! We can sometimes get weird results when we change it.
For example in an alternating series, what if we made all positive terms come first? So be careful!
There are other types of Infinite Series, and it is interesting and often challenging! Hide Ads About Ads.
Infinite Series The sum of infinite terms that follow a rule. When we have an infinite sequence of values: 1 2 , 1 4 , 1 8 , 1 16 , The sums are heading towards 1, so this series is convergent. The "sum so far" is called a partial sum.
So, more formally, we say it is a convergent series when: "the sequence of partial sums has a finite limit. It goes up and down without settling towards some value, so it is divergent.
Sum of Infinite Series
How do we know? So the harmonic series must also be divergent.