Sequence limits and the nth-term (divergence) test for series

Key idea

A sequence \((a_n)\) converges if \(\lim_{n\to\infty} a_n\) exists and is finite. Here are reliable patterns you should know:

• Rational sequences: if degrees match, the limit is the ratio of leading coefficients. For example, \[ \lim_{n\to\infty}\frac{an+b}{cn+d}=\frac{a}{c}\quad (c\neq 0). \]
• Polynomial growth: if the numerator degree is smaller than the denominator degree, the limit is \(0\). If it’s larger, the sequence typically grows without bound.
• Exponential sequences: for \(r^n\):
  • If \(|r|<1\), then \(r^n\to 0\).
  • If \(r>1\), then \(r^n\to \infty\).
  Example: \(\left(\tfrac{2}{3}\right)^n\to 0\), but \(\left(\tfrac{7}{4}\right)^n\to \infty\).

For series convergence, the first quick filter is the nth-term (divergence) test: \[ \text{If }\lim_{n\to\infty} a_n \neq 0\text{ (or does not exist), then }\sum_{n=1}^{\infty} a_n \text{ diverges.} \] Important: \(\lim a_n = 0\) is necessary for convergence, but it does not guarantee convergence.

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Summary

• Compute sequence limits using leading terms (especially for rational expressions in \(n\)).
• For series, always check \(\lim a_n\) first: if it’s not \(0\), the series diverges.

Geometric series sums and telescoping series

Key idea

A geometric series has the form \(\sum ar^n\) (or a shifted version). It converges exactly when \(|r|<1\). The must-know sums:

• \(\displaystyle \sum_{n=0}^{\infty} r^n=\frac{1}{1-r}\) for \(|r|<1\).
• \(\displaystyle \sum_{n=1}^{\infty} r^n=\frac{r}{1-r}\) for \(|r|<1\).
• Example: \(\displaystyle \sum_{n=1}^{\infty}\left(\tfrac14\right)^n=\frac{\tfrac14}{1-\tfrac14}=\frac{1}{3}\).

A telescoping series is one where many terms cancel after rewriting: \[ a_n = b_n - b_{n+1}\quad \Rightarrow\quad \sum_{n=1}^{N} a_n = b_1 - b_{N+1}. \] This often happens after partial fraction decomposition.

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Summary

• Geometric series converge when \(|r|<1\) and then you can sum them exactly.
• Telescoping series become easy after rewriting terms so they cancel in the partial sum.

p-series, the harmonic series, and quick classification

Key idea

The p-series test is one of the fastest convergence checks:

• \(\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^p}\) converges if \(p>1\).
• \(\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^p}\) diverges if \(p\le 1\).
• The harmonic series \(\sum \frac{1}{n}\) is the case \(p=1\), so it diverges.

You’ll also use p-series as benchmarks in the comparison test. If your terms behave like \(\frac{1}{n^p}\) for large \(n\), then the p-series test often tells you what happens.

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Summary

• \(\sum \frac{1}{n^p}\) converges if \(p>1\), diverges if \(p\le 1\).
• The harmonic series \(\sum \frac{1}{n}\) diverges (it’s \(p=1\)).

Comparison tests and “exponential beats polynomial”

Key idea

The comparison test is a fast strategy when terms are positive:

• If \(0\le a_n \le b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges.
• If \(0\le b_n \le a_n\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.

A powerful intuition: exponentials beat polynomials. That’s why terms like \(\dfrac{1}{n2^n}\) often converge: the \(2^n\) in the denominator shrinks the terms very fast.

Also, this identity is extremely useful for sums: \[ \sum_{n=1}^{\infty} n r^n=\frac{r}{(1-r)^2}\quad (|r|<1). \] (You can derive it by differentiating the geometric series.)

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Summary

• Comparison works best when you can sandwich your series between simple “benchmark” series.
• Know \(\sum n r^n=\dfrac{r}{(1-r)^2}\) for \(|r|<1\) to compute common convergent sums quickly.

Ratio test, root test, and factorial/exponential series

Key idea

The ratio test looks at \[ L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|. \] Then:

• If \(L<1\), the series \(\sum a_n\) converges absolutely.
• If \(L>1\) (or \(L=\infty\)), the series diverges.
• If \(L=1\), the test is inconclusive.

The root test uses \[ L=\limsup_{n\to\infty}\sqrt[n]{|a_n|}. \] It has the same conclusion rules as the ratio test and is especially nice for terms like \((\text{something})^n\).

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Summary

• Ratio/root tests are perfect for factorials and exponential/power series terms.
• For geometric sequences, \(\frac{a_{n+1}}{a_n}\) is the constant ratio \(r\).

Alternating series test and absolute vs conditional convergence

Key idea

An alternating series often looks like \[ \sum_{n=1}^{\infty} (-1)^{n-1} b_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^{n} b_n, \] where \(b_n \ge 0\). The alternating series test says the series converges if:

• \(b_n\) is eventually decreasing, and
• \(\lim_{n\to\infty} b_n=0\).

A series converges absolutely if \(\sum |a_n|\) converges. If \(\sum a_n\) converges but \(\sum |a_n|\) diverges, then \(\sum a_n\) converges conditionally.

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Summary

• Alternating series test: decreasing \(b_n\to 0\) \(\Rightarrow\) \(\sum (-1)^n b_n\) converges.
• Absolute convergence means \(\sum |a_n|\) converges; conditional means \(\sum a_n\) converges but \(\sum |a_n|\) diverges.

Power series convergence: radius and interval of convergence

Key idea

A power series centered at \(a\) looks like \[ \sum_{n=0}^{\infty} c_n (x-a)^n. \] Usually, there is a radius \(R\) such that:

• The series converges for \(|x-a|<R\).
• The series diverges for \(|x-a|>R\).
• At \(|x-a|=R\), you must test endpoints separately.

The ratio test often produces a condition like \(|x-a|<R\), and then \(R\) is the radius of convergence.

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Final recap

• Nth-term test: if \lim a_n≠ 0, then \(\sum a_n\) diverges.
• Geometric series: converge when \(|r|<1\), sum with \(\frac{1}{1-r}\) (or \(\frac{r}{1-r}\) when starting at \(n=1\)).
• Telescoping: rewrite to cancel in partial sums; take the limit.
• p-series: \(\sum \frac{1}{n^p}\) converges if \(p>1\), diverges if \(p\le 1\).
• Comparison/limit comparison: match to a known benchmark series.
• Ratio/root tests: best for factorials, exponentials, and power series.
• Alternating series: decreasing \(b_n\to 0\) implies convergence; decide absolute vs conditional by checking \(\sum |a_n|\).
• Power series: find radius \(R\) and then test endpoints to get the interval of convergence.