Home
Random
Log in
Settings
Donate
About Wikisource
Disclaimers
Search
Page
:
Elements of the Differential and Integral Calculus - Granville - Revised.djvu/250
Previous page
Next page
Page
Discussion
Image
Index
Language
Watch
Edit
This page needs to be
proofread
.
23.
cos
x
e
x
+
cos
2
x
e
2
x
+
cos
3
x
e
3
x
+
⋯
.
{\displaystyle {\frac {\cos x}{e^{x}}}+{\frac {\cos 2x}{e^{2x}}}+{\frac {\cos 3x}{e^{3x}}}+\cdots .}
Ans.
x
>
0.
{\displaystyle x>0.}
.
Hint.
Neither the sine nor cosine can exceed 1 numerically.
24.
1
+
x
log
a
+
x
2
log
2
a
2
!
+
x
3
log
3
a
2
!
+
⋯
.
{\displaystyle 1+x\log a+{\frac {x^{2}\log ^{2}a}{2!}}+{\frac {x^{3}\log ^{3}a}{2!}}+\cdots .}
Ans.
All values of
x
{\displaystyle x}
.
25.
1
1
+
x
+
1
1
+
x
2
+
1
1
+
x
3
+
⋯
.
{\displaystyle {\frac {1}{1+x}}+{\frac {1}{1+x^{2}}}+{\frac {1}{1+x^{3}}}+\cdots .}
Ans.
x
>
1.
{\displaystyle x>1.}
26.
x
+
1
2
⋅
x
3
3
+
1
2
⋅
3
4
⋅
x
5
5
+
1
2
⋅
3
4
⋅
5
6
⋅
x
7
7
+
⋯
.
{\displaystyle x+{\frac {1}{2}}\cdot {\frac {x^{3}}{3}}+{\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {x^{5}}{5}}+{\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{6}}\cdot {\frac {x^{7}}{7}}+\cdots .}
Ans.
−
1
≤
x
≤
1.
{\displaystyle -1\leq x\leq 1.}
27.
1
+
x
+
2
x
2
+
3
x
3
+
⋯
.
{\displaystyle 1+x+2x^{2}+3x^{3}+\cdots .}
28.
x
−
x
3
3
+
x
5
5
−
x
7
7
+
⋯
.
{\displaystyle x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots .}
29.
10
x
+
100
x
2
+
1000
x
3
+
⋯
.
{\displaystyle 10x+100x^{2}+1000x^{3}+\cdots .}
30.
1
+
x
+
(
2
!
)
x
2
+
(
3
!
)
x
3
+
⋯
.
{\displaystyle 1+x+(2!)x^{2}+(3!)x^{3}+\cdots .}