Find a power series for f(x)=xln(1+x). \[f(x)=xln(1+x)\]. \[\frac{d}{dx}xln(1+x)\]. \[=\frac{x}{x+1}+ln(x+1)\]. \[=\sum_{n=0}^{\infty}(-1)^nx^n+\fr ac{d}{dx}ln(x+1)\]. \[\sum_{n=0}^{\infty}(-1)^nx^n+\sum _{n=0}^{\infty}(-1)^n\frac{x^{n+1}} {n+1}\]

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.