级数收敛和函数表示

geometric series

• defination: multiply each term by a fixed numer to get the next term(等比)
• summation: $$S_n=\frac{a(1-r^n)}{1-r}$$ here\(S_n\) is the sum of n terms, a is the first term, r is the fixed number. and te sum of the series would be: $$S=lim_{n\rightarrow\infty} S_n$$
• a geometric series has a sum if and only if \(|r|<1\), and in this case the sum is $$S=\frac{a}{1-r}$$ this series is then called convergent.

some tricks

• 指数函数增长比多项式快
• 阶乘增长比指数快
• stirling公式（数学中用来近似阶乘的重要公式）$$n!\eqsim\sqrt{2\pi n}(\frac{n}{e})^n$$

convergence

• definition: if a series has a finite sum, it is called convergent. Otherwise it is called divergent.

tests for convergence

• Preliminary Test: if \(lim_{n\rightarrow\infty}a_n\neq0\) the series diverges. If \(lim_{n\rightarrow\infty}a_n \equiv 0\), we must test further.
  if a series converges absolutely then it converges
• the comparision test: make sure that all terms are positive.
  • let \(\sum_{n=0}^{\infty}m_n\) be a series of positive terms which you know converges. Then the series you are testing: \(\sum_{n=0}^{\infty}a_n\) is absolutely convergent if \(|a_n|\le m_n\) for all n from some point on.
  • let \(\sum_{n=0}^{\infty}d_n\) be a series of positive terms which you know diverges. Then the series $$\sum_{n=0}^{\infty}|a_n|$$ diverges if \(|a_n|\ge d_n\) for all n from some point on.
• the integral test: make sure that all terms are positive and not increasing if \(0<a_{n+1}\le a_n\) for \(n>N\), then \(\sum^{\infty}a_n\) converges if \(\int^{\infty}a_ndn\) is finite and diverges if the integral is infinite.(the integral is to be evaluate only at upper limit, no lower limit is needed)
• the ratio test: make sure that all terms are positive. \(\rho_n=|\frac{a_{n+1}}{a_n}|\) and \(\rho=lim_{n\rightarrow\infty}\rho_n\). if \(\rho<1\) the series converges; if\(\rho=1\) use a different test; if \(\rho>1\) the series diverges.
• the special comparison test: make sure that all terms are positive. 将\(a_n\)写成一个简洁的主导项形式，然后构造比较对象\(b_n\)，然后计算极限\(lim_{n\rightarrow\infty}\frac{a_n}{n_n}=L\)，且\(0<L<\infty\)则\(\sum a_n\)和\(\sum b_n\)同时收敛或发散。当我们构造比较对象时经常采用以下几种：
  • P级数：\(\sum \frac{1}{n^p}\) 当且仅当\(p>1\)时收敛，其他情况下级数发散。通常用integral test测试。
  • 几何级数：\(\sum ar^n\) 当且仅当 \(|r|<1\)时收敛
  • 对数级数：\(\sum \frac{1}{n ln n}\)、\(\sum\frac{1}{n(ln n)^p}\)当且仅当\(p>1\)时收敛

some result worth memorizing

• \(\sum_{n=1}^{\infty}\frac{1}{n}\) divergent

• \(\sum_{n=1}^{\infty}\frac{1}{n^2}\) convergent

• \(\sum_{n=1}^{\infty}\frac{1}{2^n}\) divergent

• \(\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\) divergent

• \(\sum_{n=1}^{\infty}\frac{1}{ln n}\) divergent

• test fo alternating series: if \(|a_{n+1}|\le|a_n|\) and \(lim_{n\rightarrow\infty}a_n=0\) the series converges.

taylor's series

\[f(x)=f(a)+(x-a)f'(a)+\frac{1}{2!}(x-a)^2f''(a)+\dots++\frac{1}{n!}(x-a)^2f^{(n)}(a) \]

\[f^{(n)}(a)=\frac{d^n}{dx^n}f(x)|_{x=a} \]