Series and sequences?

This is for self-reference ( I will be answering it ).
Feel free to edit if you want to correct anything, though.

  1. Determine if the following sequences are convergent or divergent.
    a) a_n=(2n^3-6)/(n^2+20n)
    b) a_n=\tan^-1(n)
    c) a_n=(1+n)^(1/n)
    d) a_n=(1-4/n)^n

  2. Determine if the following series are convergent or divergent. Use the appropriate test and check all the conditions.
    a) 6-2+2/3-2/9+2/27-2/81+...
    b) \sum_(n=1)^\infty1/(n(1+\ln^2n))
    c) \sum_(n=1)^\infty(7+n)/(n+6)
    d) \sum_(n=1)^\infty\ln(n)/n
    e) \sum_(n=0)^\infty(\pi/(2e))^n
    f) \sum_(n=2)^\infty1/(n(\ln(n))^2
    g) \sum_(n=1)^\infty(1-1/n)^n

  3. Find the values of x of which the following series is convergent.
    \sum_(n=0)^\infty(2x-3)^n

  4. Solve the following differential equations.
    a) y-xy'=3-2x^2y'
    b) y'-1/xy=2x^2\ln(x)

1 Answer
Apr 9, 2018
  1. a) \infty, divergent
    b) \pi/2, convergent
    c) 1, convergent \color(red)(♦)
    d) 1/e^4, convergent

  2. a) r\lt1, convergent \color(red)(♦)
    b) \pi/2, convergent
    c) 1, divergent
    d) \infty, divergent
    e) convergent to (2e)/(2e-\pi)
    f) 1/\ln(2), convergent
    g) 1/e\ne0, divergent

  3. convergent on 1\ltx\lt2 \color(red)(♦)

  4. a) y=3+x/(A(2x-1)) OR y=3x+(Cx)/(2x-1) \color(red)(♦)
    b) y=x^3\lnx-x^3/2+Cx \color(red)(♦)

Explanation:

\color(red)(♦) Some questions have been answered previously; I credit the users who helped me below, with URLs.

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