Series and sequences?

Question

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

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#

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#

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

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

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Answer 1 · 2018-04-09T06:12:35

a) #\infty#, divergent
b) #\pi/2#, convergent
c) #1#, convergent #\color(red)(♦)#
d) #1/e^4#, convergent
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
convergent on #1\ltx\lt2# #\color(red)(♦)#
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)(♦)#

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

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