Defining Convergent and Divergent Infinite Series
Introduced| English |
|---|
| infinite series/ˈɪnfɪnət ˈsɪəriːz/ |
| converges/kənˈvɜːdʒɪz/ |
| diverges/daɪˈvɜːdʒɪz/ |
| partial sum/ˈpɑːʃl sʌm/ |
Adding up infinitely many numbers
- Can you add up infinitely many terms and get a finite total? Sometimes yes.
- An infinite series 无穷级数 is the sum $a_1+a_2+a_3+\cdots$ of the terms of a sequence.
- It converges 收敛 if the running total approaches a finite number, and diverges 发散 otherwise.
- This unit is all about deciding which — and, when it converges, to what.
Partial sums are the key
- Add the terms one at a time: the partial sum 部分和 $S_n=a_1+a_2+\cdots+a_n$ is the total of the first $n$ terms.
- These partial sums form their own sequence $S_1,S_2,S_3,\dots$.
- The series' value is defined as the limit of the partial sums: $\displaystyle\sum a_n=\lim_{n\to\infty}S_n$.
- So an infinite sum is really a limit — the same limit idea from Unit 1.
Terms of a converging series
The terms of $1+\tfrac12+\tfrac14+\cdots$ shrink geometrically, so the partial sums settle at a finite limit.
The value of an infinite series is defined as the limit of its...
$\sum a_n=\lim_{n\to\infty}S_n$.
The sum of the first $n$ terms, $S_n$, is called the $n$th ____ sum.
The series is the limit of these partial sums.
Converge vs. diverge
- Converges: the partial sums $S_n$ approach a finite limit $L$ — that $L$ is the series' sum.
- Diverges: the partial sums grow without bound, or bounce and never settle.
- $1+\tfrac12+\tfrac14+\tfrac18+\cdots$ converges (to $2$); $1+1+1+\cdots$ diverges.
- Convergence is about the partial sums settling down, not the terms.
The partial sums of $1+\tfrac12+\tfrac14+\cdots$ climb toward what limit?
A geometric series with ratio $\tfrac12$ sums to $2$.
Which series clearly diverges?
Terms don't shrink to $0$, so it diverges.
A series converges when its partial sums...
Convergence = partial sums approach a finite $L$.
Terms shrinking is necessary, not enough
- For a series to converge, its terms must shrink to $0$ — otherwise the total can't settle.
- But shrinking terms alone don't guarantee convergence (the harmonic series $\sum\tfrac1n$ has terms $\to0$ yet diverges).
- So "$a_n\to0$" is a necessary condition, not a sufficient one.
- The tests in this unit are the tools for the harder cases.
If the terms $a_n\to 0$, the series must converge.
Necessary but not sufficient — e.g. $\sum\tfrac1n$ diverges.
A series is a limit of partial sums, not just "the terms." Terms going to $0$ is necessary for convergence but not sufficient — $\sum\frac1n$ has terms $\to0$ yet diverges. Don't conclude convergence just because the terms shrink; you need one of the convergence tests.
Does $1+\tfrac12+\tfrac14+\tfrac18+\cdots$ converge?
- Partial sums: $S_1=1$, $S_2=1.5$, $S_3=1.75$, $S_4=1.875,\dots$
- They climb toward $2$ and never pass it.
- The limit is $2$, so the series converges to $2$.
An infinite series $\sum a_n$ is the limit of its partial sums $S_n=a_1+\cdots+a_n$: it converges to $L$ if $S_n\to L$ (finite), and diverges otherwise. Terms shrinking to $0$ is necessary but not sufficient for convergence.