determine whether the series is absolutely convergent, conditionally convergent, or divergent sin(n)/3^n convergent

absolutely convergent

Step-by-step explanation:

given data

sin(n)/3^n

solution

we have given term

when n = 1

and we know that

value of sin(n) ≤ 1

so that we can say that

 ≤   or

here  is converges this is because common ratio in geometric series

here r is  and here it satisfy that -1 < r < 1

so it is converges

and

 is also similar

so it is  converges

and here no  term is

so we can say series is absolutely convergent

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

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step-by-step explanation: