1) Introduction to Sequences and Series 🤔
At an expert level, we explore sequences (ordered lists of numbers) and series (the sums of such lists) in more depth. You’ve seen the basics in earlier courses—arithmetic and geometric sequences, simple infinite series, etc. Now we expand to include:
- General definitions of sequences, convergence, and divergence.
- Infinite series: partial sums and tests for convergence.
- Advanced ideas: absolute versus conditional convergence and a glimpse of power series.
- Many examples to handle complex problems or prepare for calculus expansions.
We’ll start simple to refresh the basics, then ramp up to advanced topics such as convergence tests and series expansions.
2) Basics of Sequences 🔢
A sequence is a function from the natural numbers to the real (or complex) numbers:
\( a_1,\, a_2,\, a_3,\, \dots \quad \text{or} \quad (a_n)_{n=1}^{\infty} \)
Convergence: A sequence \( (a_n) \) converges to \( L \) if, for large \( n \), the terms \( a_n \) get arbitrarily close to \( L \). If no such \( L \) exists, the sequence diverges.
2.1 Simple Examples
- Arithmetic Sequence: \( a_n=a_1+(n-1)d \).
- Geometric Sequence: \( a_n=a_1\, r^{n-1} \). Converges if \(|r|<1\), diverges otherwise.
Example 1: \( a_n=\frac{1}{n} \). Then, \(\lim_{n\to\infty} a_n=0\).
Example 2: \( b_n=2^n \). Then, \(\lim_{n\to\infty} b_n=\infty\) (diverges).
3) Infinite Series: Summation, Partial Sums, Convergence ✨
A series is an infinite sum:
\( \sum_{n=1}^{\infty} a_n = a_1+a_2+a_3+\cdots \)
We define the partial sum:
\( S_N=\sum_{n=1}^{N} a_n \)
If \( \lim_{N\to\infty} S_N \) exists and is finite, we say the series converges to that limit. Otherwise, it diverges.
3.1 Examples
- Geometric Series:
\( \sum_{n=0}^{\infty} ar^n \)
Converges if \(|r|<1\) to \(\frac{a}{1-r}\), and diverges if \(|r|\ge1\). - Example 3: \(\sum_{n=0}^{\infty} \frac{1}{2^n}=2\).
- Harmonic Series:
\( \sum_{n=1}^{\infty} \frac{1}{n} \)
diverges.
4) Tests for Convergence 🧩
4.1 Basic Tests
- n-th Term Test: If \(\sum a_n\) converges, then \(\lim_{n\to\infty} a_n=0\). (However, a limit of 0 alone doesn’t guarantee convergence, as in the harmonic series.)
- Comparison Test: Compare with a series known to converge or diverge.
- Ratio Test:
\( L= \lim_{n\to\infty} \Bigl|\frac{a_{n+1}}{a_n}\Bigr| \)
If \(L<1\), the series converges; if \(L>1\), it diverges; if \(L=1\), the test is inconclusive. - Integral Test: (Requires calculus.) Use the integral of the function to determine convergence.
5) Absolute vs. Conditional Convergence ✨
A series is said to be absolutely convergent if \(\sum |a_n|\) converges. In that case, \(\sum a_n\) also converges.
If \(\sum a_n\) converges but \(\sum |a_n|\) diverges, then it is conditionally convergent.
Example 4:
- The alternating harmonic series \(\sum (-1)^n\frac{1}{n}\) converges conditionally.
- \(\sum \frac{(-1)^n}{n}\) converges, but \(\sum \frac{1}{n}\) diverges.
6) Advanced Topics & Power Series (Brief) 🔥
A power series is of the form:
\( \sum_{n=0}^{\infty} c_n (x-a)^n \)
Convergence depends on the radius \( R \). If \(|x-a| Example 5: \(\sum_{n=0}^{\infty} x^n\) converges for \(|x|<1\) and sums to \(\frac{1}{1-x}\).
We won’t cover every test (like the root test or Dirichlet test) here, but know that advanced techniques exist for deeper analysis.
7) Worked Examples (Detailed) 🍀
Example 6: Geometric vs. Non-Geometric Series
Problem: Compare \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) and \(\sum_{n=1}^{\infty} \frac{n}{2^n}\).
- \(\frac{1}{2^n}\) is geometric with ratio \(\frac{1}{2}<1\); its sum is finite.
- \(\frac{n}{2^n}\) isn’t strictly geometric, but applying the ratio test:
\( \lim_{n\to\infty}\Bigl|\frac{(n+1)/2^{n+1}}{n/2^n}\Bigr| = \frac{1}{2} \)
Since \(\frac{1}{2}<1\), the series converges.
Example 7: Alternating Series
Problem: Consider \(\sum_{n=1}^{\infty} (-1)^n\frac{1}{n^2}\). Does it converge?
Since \(\sum \frac{1}{n^2}\) converges (p-series with \(p=2>1\)), the alternating series converges absolutely.
Example 8: Conditional Convergence
Problem: Evaluate \(\sum_{n=1}^{\infty} (-1)^n\frac{1}{n}\).
Although \(\lim_{n\to\infty}\frac{1}{n}=0\), the absolute series \(\sum \frac{1}{n}\) diverges. However, the alternating harmonic series converges by the alternating series test (conditionally).
8) Practice Questions 🎯
8.1 Fundamental – Build Skills (10+)
- Find \(\lim_{n\to\infty} \frac{2n}{n+3}\) for the sequence \( a_n=\frac{2n}{n+3} \).
- An arithmetic sequence has \( a_1=7 \) and \( d=-2 \). Write the formula for \( a_n \) and find \( a_{10} \).
- If a geometric sequence has \( a_1=8 \) and \( r=0.5 \), find \( a_5 \).
- Determine whether the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) converges or diverges.
- Evaluate the geometric series \(\sum_{n=0}^{\infty} 3\Bigl(\frac{2}{3}\Bigr)^n\). Does it converge? If so, find its sum.
- If \(\sum a_n\) converges, must \( a_n \to 0 \)? Explain using an example.
- Decide whether \(\sum_{n=1}^{\infty} (-1)^n\frac{1}{n^3}\) converges absolutely or conditionally.
- Use the ratio test on \(\sum_{n=1}^{\infty} \frac{n}{3^n}\) to determine convergence.
- Compare \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) with a known p-series to explain its convergence.
- For \(\sum_{n=1}^{\infty} \frac{n}{2^n}\), calculate the partial sum up to \( n=4 \) and estimate the limit as \( n \to \infty \).
8.2 Challenging – Push Limits (5+)
- 🔥 Prove or disprove: Does \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n^{1.1}}\) converge absolutely or conditionally?
- 🔥 Explain why the p-series \(\sum \frac{1}{n^p}\) converges for \( p>1 \) and diverges for \( p\le1 \), referencing known results.
- 🔥 For the power series \(\sum_{n=0}^{\infty} x^n\), determine for which \( x \) it converges and find its sum.
- 🔥 (Optional) Apply the root test to \(\sum \frac{3^n}{4^n}\) and interpret the result.
- 🔥 In a real-world context, if an investment yields \(\frac{1}{n^2}\) dollars each year, does the total payout converge? (Hint: \(\sum_{n=1}^{\infty} \frac{1}{n^2}\approx \frac{\pi^2}{6}\).)
9) Summary
- Sequences are analyzed by examining their limits to determine convergence or divergence.
- Infinite series are defined via partial sums; convergence is determined by various tests (n-th term, comparison, ratio, integral, etc.).
- Absolute convergence implies convergence; conditional convergence occurs when the series converges but its absolute series diverges.
- Advanced topics like power series and p-series set the stage for calculus and further analysis.
Mastery of sequences and series is fundamental for further studies in calculus, real analysis, finance, and physics. Keep practicing and exploring these infinite processes! 🌟