sqrt sin(30)+sin(45)+sin(60)+sin(75).....infnity

since sin(180+x) = -sin(x) this series cannot be evaluated. Or, it can be evaluated in a variety of different ways, giving different answers.

The ratio test fails.

first assumption: the square root is over the whole thing

2nd:
it would be nice if the series has included the terms
sin 0 and sin 15
well, let's put them in

so we have
√ ( -sin 0 - sin 15 + (sin 0 + sin15 + sin30 + ... sin 345) + (sin 360 + sin 375 + .... sin 705) + (......) )

as Steve noted , we have opposite terms in each of the set of brackets
e.g.
sin 0 + sin 15 + .... + sin 345) = 0
so is each of the rest of the brackets
since the series goes to infinity,
I see the series as
√( 0 - sin 15 + 0+0+0+....)
which of course is not a real number, since -sin15° is negative,
and as Steve said , it cannot be evaluated.

To calculate the value of the infinite series √sin(30°) + sin(45°) + sin(60°) + sin(75°) + ..., we need to use a mathematical technique called infinite series summation. This particular series seems to be an arithmetic progression of sines of angles.

Let's break down the steps to calculate the sum:

Step 1: Write down the general term of the series.
Looking at the pattern, it seems that each term can be written as the sine of an angle that increases by 15 degrees for each subsequent term. Therefore, the nth term can be written as sin(15n), where n is the position of the term in the series.

Step 2: Derive a formula for the partial sum of the series.
To find the sum of an infinite series, we first calculate the partial sum of the first n terms. Let's denote this partial sum by S(n).

S(n) = sin(15) + sin(30) + sin(45) + ... + sin(15n)

Step 3: Simplify and transform the partial sum.
We can use the trigonometric identity sin(a) = 2sin(a/2)cos(a/2) to express sin(15) as a product of sines of smaller angles.

S(n) = 2sin(15/2)cos(15/2) + 2sin(30/2)cos(30/2) + 2sin(45/2)cos(45/2) + ... + 2sin(15n/2)cos(15n/2)

Step 4: Apply the formula for the sum of a finite geometric series.
We can see that each term in the above expression can be written as sin(k/2) * cos(k/2), where k is a multiple of 15. These terms form a geometric series, with the common ratio being cos(15/2).

Using the formula for the sum of a finite geometric series, the partial sum S(n) can be expressed as:

S(n) = 2sin(15/2) [ (1 - (cos(15/2))^n ) / (1 - cos(15/2)) ]

Step 5: Find the limit as n approaches infinity.
To find the sum of an infinite series, we need to find the limit of the partial sum S(n) as n approaches infinity:

S(infinity) = lim(n → infinity) S(n)

By evaluating this limit, we can find the sum.

Note: Evaluating this limit analytically can be quite challenging, and it might not be possible to obtain a simple closed-form expression for the sum. In such cases, numerical methods or approximation techniques may be necessary to calculate the sum.

Please note that this process may not lead to an exact answer due to the nature of the series. It is always useful to apply numerical methods or approximation techniques if an exact expression is not achievable.