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I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I. Alternately, see Zygmund's original paper.

I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I. Alternately, see Zygmund's original paper.

I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I.

Added. I am not so sure that a simple solution for part (b) exists, given that it implies Theorem B in Zygmund's original paper, which is already rather deep. (The proof takes 3 pages, and in fact Zygmund remarks in the beginning of Section 2 that for the existence of radial limits "the proof follows the same lines and may be left to the reader".) This paper is also the original source for the result I quoted from his book, see Theorem D there.

I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I. Alternately, see Zygmund's original paper.

I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I.

Added. I am not so sure that a simple solution for part (b) exists, given that it implies Theorem B in Zygmund's original paper, which is already rather deep. (The proof takes 3 pages, and Zygmund remarks in the beginning of Section 2 that for the existence of radial limits "the proof follows the same lines and may be left to the reader".) This paper is also the original source for the result I quoted from his book, see Theorem D there.

I don't know any simply way, but I would be interested in one, too.

In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.

For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I.

Added. I am not so sure that a simple solution for part (b) exists, given that it implies Theorem B in Zygmund's original paper, which is already rather deep. (The proof takes 3 pages, and in fact Zygmund remarks in the beginning of Section 2 that for the existence of radial limits "the proof follows the same lines and may be left to the reader".) This paper is also the original source for the result I quoted from his book, see Theorem D there.