I suppose you mean

Recall that

which converges everywhere. Then by substitution,

which also converges everywhere (and we can confirm this via the ratio test, for instance).
a. Differentiating the Taylor series gives

(starting at
because the summand is 0 when
)
b. Naturally, the differentiated series represents

To see this, recalling the series for
, we know

Multiplying by
gives

and from here,


c. This series also converges everywhere. By the ratio test, the series converges if

The limit is 0, so any choice of
satisfies the convergence condition.
<span>5a + 2 = 6 - 7a
5a + 7a = 6 - 2
12a = 4
a = 4/12 = 1/3
</span>
Answer:
10.5 miles
Step-by-step explanation:
Let b represent the distance traveled by boat. The relation between speed, distance, and time is ...
time = distance / speed
We are told the total time and the total distance so we have ...
b/7 + (14-b)/2 = 3.25
2b +7·14 -7·b = 14·3.25 . . . multiply by 14
3.75·14 = 5b . . . . . . . . . . . .add 5b-14·3.25
b = 10.5 . . . . . . . . . . . . . . . divide by 5
They traveled 10.5 miles by boat.
_____
<em>Check</em>
10.5/7 +3.5/2 = 1.5 +1.75 = 3.25 . . . . the answer checks OK
They spent 1.75 hours walking 3.5 miles to the dock and 1.5 hours traveling 10.5 miles by boat.
2.1 x 10 to the power of 4.
10x10= 100 x 10= 1,000 x 10= 10,000
10,000 x 2.1= 21,000