Hey guys, I have a few problems that if possible, I would love to have someone check my solutions to them.
Problem 1. Determine whether the improper integral
 dx)
converges, or diverges. In the case of convergence, give its value.
My Solution ![= \lim_{t \to \infty}\int^{t}_{1}(\frac{1}{x} - \frac{2x}{x^2 + 1}) dx = \lim_{t \to \infty}[lnx - ln(x^2 + 1)]^{t}_{1} = \lim_{t \to \infty}[(lnt - ln(t^2 + 1) - (ln1 - ln2)] =](http://latex.codecogs.com/png.latex?= \lim_{t \to \infty}\int^{t}_{1}(\frac{1}{x} - \frac{2x}{x^2 + 1}) dx = \lim_{t \to \infty}[lnx - ln(x^2 + 1)]^{t}_{1} = \lim_{t \to \infty}[(lnt - ln(t^2 + 1) - (ln1 - ln2)] =)
 + ln2) = \infty - \infty + ln2 = ln2)
... Converges to ln2
Nop: ![\ln t-\ln(t^2+1)=\ln\left(\frac{t}{t^2+1}\right)\xrighta rrow[t\to\infty]{}-\infty](http://latex.codecogs.com/png.latex?\ln t-\ln(t^2+1)=\ln\left(\frac{t}{t^2+1}\right)\xrighta rrow[t\to\infty]{}-\infty)
and thus the integral diverges. Of course, you can't write 
... Problem 2. Determine whether the sequence with the given general term converges, or diverges. In the case of convergence, give its value.
a.) 
My Solution 
, Using L'Hopital's Rule.
... Converges to
Nop. You can't use DIRECTLY L'Hospital with a discrete variable since L'H implies the use of derivative which use limits which need a continuous variable. Of course, you can use L'H with 
and then use that the limit stays the same no matter how 
, and thus this is so if you choose to go to the limit along the naturals. Another way, perhaps more natural , to divide both numerator and denominator by the highest power of n and use arithmetic of limits: ![\frac{2n-1}{3-5n}\cdot\frac{1\slash n}{1\slash n} = \frac{2-\frac{1}{n}}{\frac{3}{n}-5}\xrightarrow [n\to\infty]{}\frac{2-0}{0-5}=-\frac{2}{5}](http://latex.codecogs.com/png.latex?\frac{2n-1}{3-5n}\cdot\frac{1\slash n}{1\slash n} = \frac{2-\frac{1}{n}}{\frac{3}{n}-5}\xrightarrow [n\to\infty]{}\frac{2-0}{0-5}=-\frac{2}{5})
Tonio b.) ^{n}\sin{\frac{n\pi}{2}})
My Solution ^{n}\sin{\frac{n\pi}{2}}| = \lim_{n \to \infty}|(-1)^{n}|*|\sin{\frac{n\pi}{2}}| = \lim_{n \to \infty}1 * \sin{\frac{n\pi}{2}} =)
No Limit, Diverges
c.) ^n)
My Solution 
...
^{n}, lny = nln(1 + \frac{1}{n}))
}{\frac{1}{n}} = \frac{0}{0},)
Using L'Hopital's Rule.
...
... Converges to e
Problem 3. Calculate the sum of the given convergent geometric series.
a.) 
...
My Solution 
,
b.) ^{n}(\frac{3}{7})^n)
My Solution 
,
Problem 4. Write the first three terms of the sequence of partial sums of the series
.
My Solution 
,
,
Thank you in advance for any help!
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Originally Posted by mturner07 
